SCHOLAR Study Guide CfE Higher Mathematics Course Materials Topic : Vectors Authored by: Margaret Ferguson Reviewed by: Jillian Hornby Previously authored by: Jane S Paterson Dorothy A Watson Heriot-Watt University Edinburgh EH4 4AS, United Kingdom.
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Topic Vectors Contents. Looking back at vectors from National....................... Vector Journeys............................... 4.. Add, Subtract & Multiply Vectors in D and D.............. 9.. Magnitude of a vector............................ 4. Position, unit and zero vectors........................... 7.. Position vectors................................ 7.. Unit vectors.................................... Zero vectors.................................. Parallel vectors and collinearity.......................... 4.. Parallel vectors................................ 4.. Collinearity...................................4 Division of vectors in a given ratio......................... 8. The scalar product................................... The scalar product: Component form..................... The scalar product: Geometric form...................... Perpendicular vectors............................ 4. Learning Points................................... 4.7 End of topic test................................... 44
TOPIC. VECTORS Learning objectives By the end of this topic, you should be able to: work with vectors in and dimensions; identify and use the components of a position vector; use and interpret unit vector form; use and interpret a zero vector; use and identify parallel vectors; determine collinearity of vectors; determine the coordinates of a point that divides a line internally; calculate the scalar product; determine the angle between two vectors; identify perpendicular vectors.
TOPIC. VECTORS. Looking back at vectors from National Summary A vector is a quantity which has both direction and magnitude. The magnitude of a vector is its size or length. A directed line segment from A to B is defined as AB. A vector or force can also be defined by a lower-case letter in bold. Vectors are equal if they have the same direction and magnitude. When a vector has its direction reversed it is negative e.g. a becomes -a. A vector can be multiplied by a scalar e.g. doubling a gives a. Vectors can be added by joining one to the end of another. Displacement is the shortest distance from A to B. A vector journey is a description of its displacement. The components of a vector describe the journey from A to B e.g. x y in D. z ( x y ) in D or ( Arithmetic ) can ( be ) performed ( on the ) components ( ) (+ ) e.g. 4 + 4 + = = + The magnitude is calculated from the components using the Theorem of ( ) x Pythagoras e.g. u =, u = x x + y and v = y, y z v = x + y + z Key point A vector is a quantity which has both direction and magnitude (or size).
4 TOPIC. VECTORS.. Vector Journeys Examples. If B is km East of A then AB is a vector with distance or magnitude km and direction East.. PQ and MN are vectors. A vector can be represented by a directed line segment, that is a line from one point to another which has a direction arrow on it. The two lines MN and PQ represent vectors in two dimensions. MN and PQ are directed line segments and the arrowheads indicate the direction.. u and v are vectors. Vectors are often represented using a lower case letter in bold.
TOPIC. VECTORS 4. Equal vectors FE and u have the same length or magnitude and are parallel with the same direction. So we can say that FE = u.. Multiplying a vector by a scalar PQand v are parallel with the same direction but PQ is double the size of v. Sowecan say that PQ = v or v = PQ.. Reversing the direction of a vector KL and a have the same magnitude and are parallel but are pointing in opposite directions. So we can say that KL = a or a = KL.
TOPIC. VECTORS Key point A displacement is the shortest distance from one point to another and a Vector Journey is a displacement. Examples. Problem: Express each of the following displacements in terms of vectors a and b. a) AC b) CB c) CA d) BC e) AB Solution: a) AC = a b) CB = b c) CA is just AC with its direction reversed so CA = a. d) BC is just CB with its direction reversed so BC = b. e) AB is the shortest distance from A to B but the long way would be from A to C to B which is AC + CB so AB = a + b. Problem: In the diagram AB = DC and BC = v. Express the displacements in terms of vectors u and v.
TOPIC. VECTORS 7 a) AB b) CD c) AD If BC = BE. Express the displacements listed as d) and e) in terms of vectors u and v. d) CE e) ED Solution: a) Since AB = DC it follows that AB and DC have the same direction, so AB = DC = u. b) CD is just DC with its direction reversed so CD = u. c) AD = AB + BC + CD = u + v + ( - u) = u + v d) Sometimes we may have to employ a little bit of algebra to find the solution. If BC = BE then y = BE. If we change the subject of the formula to BE we get y = BE. BC = BE so it follows that CE = BE. If we substitute v for BE we get CE = v = v = v. e) ED = EC + CD. If we reverse the direction of CE and DC we get ED = v + ( - u) or ED = - v u. - Vector journeys exercise Q: Express each of the following displacements in terms of vectors a and b. Go online
8 TOPIC. VECTORS a) AC b) BC c) CA d) CB e) AB f) BA Q: EFGH is a rhombus. Express each of the following displacements in terms of vectors e and f. a) EF b) HG c) EG d) HF Q: In the diagram ST = 4RQ. Express the displacements listed as a) to c) in terms of vectors w and x. If RS = 4RP. Express the displacements listed as d) to g) in terms of vectors w and x. a) RS b) ST c) TQ d) SP e) PS f) PT g) PQ
TOPIC. VECTORS 9.. Add, Subtract & Multiply Vectors in D and D Key point A vector is a quantity which has both direction and magnitude. The components of a vector describe how to get from one end of the vector to the other following its direction. Vector components in D The components ( of ) a vector are written vertically in brackets. graph 4 AB =. In the following Go online ( ) 4 a = along 4 then up describes the journey from A to B. ( ) In this graph PQ =. u = ( ) along to the left then down describes the journey from P to Q. We can add the vectors u and a together( and identify ) the components of the resultant vector. In the following graph u + a =.
TOPIC. VECTORS Notice that if( we add) the components ( ) ( of u and a we ) get the ( same) answer. 4 + 4 u + a = + = = + We can subtract the vector ( u from ) a and identify the components of the resultant vector. 7 In this graph a u =. 9 Notice that if( we subtract ) ( the components ) ( of u from a) we get ( the ) same answer. 4 4 ( ) 7 a u = = = ( ) 9 We can scale a( vector ) and identify( the components ) of the resultant vector. Here was 4 8 can see a = and a =.
TOPIC. VECTORS Notice that if we ( multiply ) the ( components ) of( a by) we get the same answer. a = 4 4 8 = = ( ) Q4: Draw a representation of the vector e = 4 7. Vector components in D The components of a vector in D are harder to visualize but the process is just the same as in D. The following graph shows AB =. Go online a = along, back then up describes the journey from A to B. We can scale a vector and identify the components of the resultant vector. Remember
TOPIC. VECTORS a =? This vector can be scaled to a = 4. Notice that if we multiply the components of a by we get the same answer. a = = = 4 Let s add three dimensional vectors. Two forces have components u = and 4 w = 7. Calculate u + w. 4 So, u + w = + 7 =. This time let s subtract three dimensional vectors. Two forces have components m = and n = 7. Calculate m - n. 8 9 So, m n = 7 =. 8 9 Now let us add, subtract and multiply three dimensional vectors. Three forces have components a = 4, b = and c =. Calculate a + b - c.
TOPIC. VECTORS a + b c = 4 + multiply the components of b by = 4 + calculate the resultant vector = Add, subtract and multiply vectors in D and D exercise ( ) ( ) ( ) 4 7 Q: a =, b = and c = 9 4 Go online a) What are the components of a + b? b) What are the components of a - b? c) What are the components of a - c? d) What are the components of 4a + b -c? 4 Q: Three forces have components a =, b = and c = 7 9 Calculate the components of the resultant vectors.. a + b. b - c. 4c 4. a - b. a -b + c
4 TOPIC. VECTORS.. Magnitude of a vector Key point The magnitude of a vector is simply its size or length, in the case of a directed line segment it s length. We can apply the Theorem of Pythagoras to find the magnitude of a vector. Go online The magnitude of a vector ( Here we have 4 AB = a = ). We have formed a right-angled triangle. The notation for magnitude is to place the name of the vector or directed line segment between vertical lines. For example, AB = a = 4 + = Calculate the magnitude of a + b. ( ) a + b = - a + b = + ( - ) = 4 or 7 We could also calculate the magnitude of a + b using their components
TOPIC. VECTORS ( ) ( ) ( ) 4 a + b = + = - - a + b = + ( - ) = 4 or 7 The Theorem of Pythagoras can be used to find magnitude of a vector in D. Here we have AB =. If we apply Pythagoras Theorem twice we can find. AB The first application of the Theorem of Pythagoras. AC = + = 9
TOPIC. VECTORS The second application of the Theorem of Pythagoras uses the length of AC. AB = ( 9 ) + = or hence AB = or There is a quicker way... AB = so, AB = + + = or Magnitude of vectors exercise Magnitude of vectors in D Go online Q7: Three forces have components a = ( ) Calculate the magnitude of the following: ( 4 ), b = ( ) and c = a) a b) b c) c
TOPIC. VECTORS 7 d) a + b e) b c f) a b + c Magnitude of vectors in D Q8: Three forces have components u = 4 Calculate the magnitude of the following: a) u b) v c) w d) u + w e) v u f) v w + u 4, v = and w =. Position, unit and zero vectors The subsequent topics cover position, unit and zero vectors. There will be opportunities to test your learning at the end of each topic... Position vectors OH is the position vector of the point H relative to the origin and is written as h.
8 TOPIC. VECTORS OG is the position vector of the point G relative to the origin and is written as g. GH can be expressed in terms of the position vectors g and h GH = ( g) +h = h g Key point AB = b a Position vectors Go online
TOPIC. VECTORS 9 Key point The components of a position vector are the same as the coordinates of the point. Examples. Problem: If M is the point (,-), what are the components of position vector m? Solution: ( ) m =. Problem: How do we express EF in terms of the position vectors e and f? Solution: EF = f e. Problem: If A(, 9) and B(,,-), find the components of AB. Solution: Position vector a has components and position vector b has components 9.
TOPIC. VECTORS AB = b a = 4 = 9 4. Problem: If C(,-,) and D(9,,4), find CD. Solution: CD = d c = = 9 4 CD = + + = + + 9 = 7 Position vectors exercise Go online Q9: P is the point (7,-) and Q is the point (,). a) What are the components of position vector p? b) What are the components of position vector q? c) What are the components PQ? d) What is PQ as a surd? Q: E is the point (,-,) and F is the point (-,, ). a) What are the components of position vector e? b) What are the components of position vector f? c) What are the components EF? d) What is EF as a surd in its simplest form?
TOPIC. VECTORS Q: If R(,-,) and T(,-,), find the components of RT. Q: If U(-,,) and V(-,,), find UV giving your answer as a surd. Q: If W(8,7,-4) and Z(,,-), find WZ giving your answer as a surd in its simplest form... Unit vectors i =, j = and k = are unit vectors. Key point A unit vector has magnitude. M is the point (-,,) and can be expressed as position vector m with components but also unit vectors. - m = and m = i +j +k. Examples. Problem: Position vector n = i 4j + k, what are the coordinates of the point N? Solution: n = 4 so N(,-4,).. Problem: Express a = Solution: a = i +j 8k in terms of the unit vectors i, j and k. 8
TOPIC. VECTORS. Problem: Find the components of the unit vector x which is parallel to y, if y = 4. Solution: Vectors x and y are parallel if x = ky i.e. x is a scaled version of y with the same direction but different magnitude. If x is a unit vector it must have magnitude. y = + ( 4) + = = of = so x = y = 4 = - 4 Unit vectors exercise Go online Q4: Add the two vectors a = and b = 4. 8 Give your answers in terms of unit vectors i, j, k. 9 Q: Subtract the vector c = 7 and d =. 4 Give your answers in terms of the unit vectors i, j, k.
TOPIC. VECTORS Q: a) Find the components of the unit vector e which is parallel to f, iff = b) What is the answer in unit vector form.. 8 Q7: If t is a unit vector and has components 7 y 7, what is a possible value of y?.. Zero vectors Key point ( The zero vector has components. ) in D or in D and is written as For any vector x, x +(-x) =. Example Problem: a If x = b, what is x +(-x)? c Solution: a If x = b c a a b + b c c then x = = a b c =
4 TOPIC. VECTORS Zero vectors practice 4 Go online Q8: a = b = 4 c = 4 What is a + b +c?. Parallel vectors and collinearity The subsequent topics cover parallel vectors and collinearity. There will be opportunities to test your learning at the end of each topic... Parallel vectors Key point If two vectors are parallel they have the same direction but their magnitudes are scalar multiples of each other. a = b tells us that the vectors are parallel and have the same magnitude. e = / f tells us that vectors e and f are parallel and that the components of e are half those of f. Examples. Problem: - Show that the two vectors a = and b = - - Solution: - If we factorise b we get - which gives b =-a. The aim when we factorise is to end up with the components of a. are parallel.
TOPIC. VECTORS. Problem: Find a vector parallel to the position vector through the point (, 4, ) with z component equal to. Solution: x Let the vector we are looking for be y. x If the vectors are parallel then y = k 4. x x 4 Since = k we find k = giving y = 4 so y = 8. The vector parallel to the position vector through the point (, 4, ) with z component 4 equal to is 8.. Problem: Are the vectors p = i j +4kand q = i 8j +kparallel? Solution: p = = 4 and q = 8 = 4 4 8 8 Since p and q are both scalars of 4 they are parallel. 8 You may also have spotted that p = q or q = p Parallel vectors practice 9 Q9: Are the vectors s = and t = parallel? Go online Q: Vectors v = i +4j +8kand w = i pj +kare parallel. What is the value of p? This use of parallel vectors can be taken a stage further in order to determine collinearity.
TOPIC. VECTORS Parallel vectors exercise Go online Q: Are the vectors i +j +kand i +4j +8kparallel? Q: Find a vector parallel to the vector 7i +j +kwhich has a y-component equal to -. Express your answer in unit vector form (e.g. ai + bj + ck). Q: Find a vector parallel to the vector a = a) State the x-component of this parallel vector. b) State the y-component of this parallel vector. c) State the z-component of this parallel vector. Q4: Are the vectors i +4j +kand i +8j +kparallel? 8 7 with z-component equal to 49. Q: Find a vector parallel to the vector i +j +kwhich has a y-component equal to -. Express your answer in unit vector form (e.g. ai + bj + ck)... Collinearity Key point Collinearity All points which lie on a straight line are said to be collinear. A,B&Carecollinear if AB andbc are parallel and the following two vector conditions can be determined: k is a scalar such that AB = kbc the point B is common to both vectors Examples. Problem: Determine whether the points A (,, ), B (,, 4) and C (9, 4, 7) are collinear. Solution: AB = b a = 4 =
TOPIC. VECTORS 7 9 BC = c b = 4 7 4 = 9 = AB = BC so BC and AB are parallel but B is a common point A, B and C are collinear. Since the point B is common to both vectors, the three points are collinear.. Problem: Show that the points A(-,-,4), B(,,7) and C(7,7,) are collinear. Solution: AB = b a = 7 4 = 7 BC = c b = 7 7 = = BC = AB so BC and AB are parallel but B is a common point A, B and C are collinear. Collinearity practice Q: Show that P(-,,-), Q(,,) and R(7,7,8) are collinear. Go online Q7: The points E(,a,7), F(,,) and G(,,) are collinear. What is the value of a? Collinearity exercise Q8: Are the points A(4, 9, 9), B -7,, -) and C(-7, -9, -89) collinear? Choose the conditions for collinearity of the points ABC. Go online
8 TOPIC. VECTORS. B is a common point. B is the midpoint of AC. There is a common point between the two vectors chosen 4. Two vectors are parallel Are the points collinear? Q9: P(-,,-9), Q(,8,-) and R(x,,9) are collinear. What is the value of x?.4 Division of vectors in a given ratio Examples. Problem: A is the point (,4,-7) and B is the point (8,,-). If M is the midpoint of AB, find the coordinates of M. Solution: 8 AB = b a = 4 = 8 7 8 If M is the midpoint then AM = AB. m a = 8 = 4 8 4 m = 4 + a = 4 + 4 4 4 7 m = 8 Since we know the components of position vector m, the coordinates of M are (,8,-).
TOPIC. VECTORS 9. Problem: A and B have coordinates (-7,4,7) and (,,4) respectively. Find the point P if it divides AB in the ratio :. Solution: 7 8 AB = b a = 4 = 4 4 7 7 Pis / 7 of the way from A to B so AP = 7AB. p a = 8 8 4 = 4 7 7 8 8 7 p = 4 + a = 4 + 4 7 p = 9 Since we know the components of position vector p, the coordinates of P are (,,9). There are alternative methods including the section formula. Key point The section formula states that if p is the position vector of the point P which n divides AB in the ratio m:n then p = m + n a + m m + n b
TOPIC. VECTORS Examples. Problem: If R is (,,-4), S(,-7,) and P divides RS in the ratio :, find the coordinates of P. Solution: Using the section formula m =, n = and m + n =. p = r + s = - 4 = = - - + - 7 = 9 9 - + - 4 So P is the point (,-,).. Problem: K is the point (-,,) and L is the point (9,,). Find the coordinates of D if KD DL =. Solution: KD DL = cross multiply to KD = DL. KD = DL (d k) = (l d) d k = l d d = k + l 7 d = + d = d = So D is the point (,,)
TOPIC. VECTORS. Problem: P, Q and R have coordinates (8,-8,-), (-,-,-) and (-4,,) respectively. If P, Q and R are collinear find the ratio in which Q divides PR. Solution: 8 PQ = q p = 8 9 = = 4 4 QR = r q = = 4 = 4 So Q divides PR in the ratio :. Division of vectors in a given ratio exercise Q: Find the position vector of the point P which divides the line AB in the ratio : and where A(,-,-4) and B(-,-,-4). a) State the x-coordinate of the point P. b) State the y-coordinate of the point P. c) State the z-coordinate of the point P. Go online Q: Find the coordinates of the point T which divides the line AB in the ratio : and where A(-,-4,-) and B(-,-,-). a) State the x-coordinate of the point T. b) State the y-coordinate of the point T. c) State the z-coordinate of the point T.
TOPIC. VECTORS Q: Find the coordinates of the point D which divides the line AB in the ratio 4: and where A(4,4,) and B(4,,4). a) State the x-coordinate of the point D. b) State the y-coordinate of the point D. c) State the z-coordinate of the point D. Q: U(-4,,), V(-,7,) and W(,9,) are collinear. What is the ratio in which V divides UW. Q4: F is the point (,,) and H(,8,). If F, G and H are collinear and FG GH =, state the coordinates of G.. The scalar product There are two ways of expressing the scalar product. The first is algebraically in component form, the second is in geometric form. The subsequent topics cover scalar products in both component and geometric form, and perpendicular vectors. There will be opportunities to test your learning at the end of each topic... The scalar product: Component form Key point Scalar product ( in ) component form ( () dimensions) If p = a d and q = b e then the scalar product is the number p q = ad + be Key point Scalar product in component form ( dimensions) a d If p = b and q = e then the scalar product is the number c f p q = ad + be + cf It is important to note that the scalar product of two vectors is not a vector. Itisa scalar. The scalar product is also known as the dot product.
TOPIC. VECTORS Examples. Scalar product (two dimensions) Problem: ( ) ( Find the dot or scalar product of a = and b = Solution: a b = ( - - ) + ( - ) = 4 + ( - ) = ).. Scalar product (three dimensions) Problem: Find the scalar product of a = and b =. Solution: a b = + + = 4. Problem: K is the point (-,-,), L(8,4,-) and M(,4). If p = LK and q = LM calculate p q. Solution: p = - 8 LK = k l = - 4 - q = 8 LM = m l = 4 4 - = = - 9-7 - 8 - p q = - 9 8 + - - + 7 = 7 + 8 + 4 =
4 TOPIC. VECTORS Top tip Algebraic rules of scalar products There are several useful properties of scalar products.. a (b + c) = a b + a c. a b = b a. a a = a Examples. Problem: u = i j +k, v = i k and w = 4j + k What is the value of u (v + w)? Solution: u =, v = and w = 4 u (v + w) = u v + u w u v = + ( - ) + ( - ) = + + ( - ) = u w = + ( - ) 4 + = + ( - ) + = u (v + w) = u v + u w = + ( - ) = - 7. Problem: If f =, prove that f f = f Solution: f f = + ( - ) ( - ) + = + ( - ) + = 4 ) f = ( + ( - ) + = + ( - ) + = 4 Hence f f = f.
TOPIC. VECTORS The scalar product: Component form practice Q: Show that the property a (b + c) = a b + a c holds for the three vectors a = i k, b = i j +kand c = i +j k Q: Using a = a a and b = b b prove a b = b a Go online a b The scalar product: Component form exercise Q7: Find the scalar product a b of the following vectors. a) When A(-,,-8) and B(4,-,-). b) For a = 8 and b =. 8 c) When A(-,9,-8) and B(-,,) Go online Q8: If a = 9 4 and b =. What is the scalar product a b? Q9: If a = 7i 7j + 4k and b = i + j k. What is the scalar product a b? Q4: P is the point (,,4), Q(7,-,) and R(,,). If a = PQ and b = PR calculate a b.
TOPIC. VECTORS.. The scalar product: Geometric form Key point The scalar product in geometric form (for an angle) The scalar product of two vectors a and b is defined as a b = a b cos θ where θ is the angle between a and b, θ 8. Notice that the vectors a and b project away for the vertex. Examples. Problem: Find the scalar product of the vectors a and b where the length of a is, the length of b is 4 and the angle between them is. Solution: a b = a b cos θ = 4 cos =. Problem: ABCD is a square of side units. If AB = a and AC = b, what is the exact value of a b? Solution: a = b = = 8 = θ = 4 a b = a b cos θ = 8 cos 4 = = 4
TOPIC. VECTORS 7. Problem: D is the point (,,), E(-,-,4) and F(,-4,-). Find the size of angle ABC. Solution: To find the angle we need the scalar product for an angle. a b = a b cos θ a = - 7 ED = d e = - = 4 4 - b = - 8 EF = f e = - 4 - = - - 4-7 a b = 7 8 + 4 ( ) + ( ) ( 7) = a = 7 + 4 + ( ) = 74 b = 8 + ( ) + ( 7) = Now let s substitute what we know into a b = a b cos θ a b = a b cos θ = 74 cos θ 74 = cos θ cos 74 = θ θ = 4 8 4. Problem: a = 4 and b =
8 TOPIC. VECTORS Find the size of θ. Solution: To find θ we need the scalar product for an angle but notice that the vectors do not project outwards from the vertex. We can rectify this by moving vector b. a b = a b cos α a b = ( - ) + 4 ( - ) + = - 9 a = ( ) + 4 + = 8 b = + ( ) + = 8 Now let s substitute what we know into a b = a b cos α a b = a b cos α - 9 = 8 8 cos α - 9 = cos α 8 8 cos - 9 8 8 = α α = So, θ = 8. = 9.9 (supplementary angles). The scalar product: Geometric form exercise Go online Q4: Two vectors a and b whose lengths are. and respectively and have an angle of between them. What is the scalar product a b? Q4: Two vectors c and d whose lengths are and 4 respectively and have an angle of 4 between them. What is the scalar product c d?
TOPIC. VECTORS 9 Q4: Given two position vectors A(,-,-) and B(4,4,7). a) What is the value of the scalar product a b? b) What is the angle, in degrees, between the two vectors a and b? Give your answer to the nearest integer. Q44: Find the angle BAC if A(-7,-8,), B(8,,-) and C(4,,8). Q4: The triangle is equilateral of side units. What is the value of e f? (Try to do this question without a calculator if you can.) Q4: What are the coordinates of G? Q47: What are the coordinates of B? Q48: What are the components of GB? Q49: What are the components of GD? Q: What is the size of the angle BGD?
4 TOPIC. VECTORS.. Perpendicular vectors Example Problem: Find the angle between i +4jand 8i +j Solution: Let a = i + 4j = 4 and b = - 8i + j = - 8. a b = 8 + 8 = a = 7 b = 8 a b = a b cos θ cos θ = a b a b = 7 8 = so θ = 9 or π The last example demonstrates an important geometric property of the scalar product. Key point If a b = then vectors a and b are perpendicular. Examples. Perpendicular vectors in component form Problem: Show that the following two vectors are perpendicular a =. and b = Solution: a b = + 4 + = The vectors are therefore perpendicular.
TOPIC. VECTORS 4. Problem: Show that the two vectors a = i +j + k and b = i +j +kare perpendicular. Solution: a b = ( ) + ( ) + ( ) = The vectors are perpendicular.. Problem: The vectors m = y and n = What is the value of y? Solution: For perpendicular vectors m n = 8 are perpendicular. ( - ) + y + 8 = - + y + 8 = y + = y = - y = - Perpendicular vectors exercise Q: What is the value of a which makes the vectors ai +4j k and 8i 9j +4k perpendicular? Go online Q: What is the value of b which makes the vectors i + bj k and i 9j +4k perpendicular? Q: What is the value of a which makes the position vectors (a, 4, -) and (-4, -, 8) perpendicular?
4 TOPIC. VECTORS. Learning Points Vector definitions A vector is a quantity which has both direction and magnitude. The magnitude of a vector is its size or length. A directed line segment from A to B is defined as AB. A vector or force can also be defined by a lowercase letter in bold. Displacement is the shortest distance from A to B. A vector journey is a description of its displacement. i =, j = and k = are unit vectors. A unit vector has magnitude. The zero vector has components ( ) in D or in D and is written as. Points which are collinear lie on the same straight line. Vectors are parallel if they have the same direction and one is a scalar multiple of the other e.g. e = f tells us that vectors e and f are parallel and that the components of e are half those of f. The scalar product is not a vector it is a scalar. Vector calculations The components of a vector describe the journey from A to B ( ) x x e.g. in D or in y D. y z Arithmetic ( ) can be ( performed ) ( on the components ) ( (+)- ) 4 + 4 e.g. + = = + The magnitude is calculated from the components using Pythagoras Theorem e.g. ( ) x u =, u = x + y y x v = y, v = x + y + z z
TOPIC. VECTORS 4 The components of a position vector are the same as the coordinates of the point. e.g. A(,,) then a =. AB = b a P, Q and R are collinear if you can show that: PQ = k QR (i.e. QR and PQ are parallel) and Q is a common point. If the point P divides AB in the ratio m:n then: n p = m + n a + m m + nb by the section formula. m Alternatively if P is m AP = AB m + n m + n of the way from A to B then: The scalar product in component form is defined as: for a = x y and b = x y z a b =x x +y y +z z z The scalar product for an angle is defined as: a b = a b cosθ, where a and b project outwards from the vertex of the angle θ and θ 8 cos θ = a b a b Properties of the scalar product a (b + c) = a b + a c a b = b a a a = a Vectors are perpendicular if a b = ; θ = 9
44 TOPIC. VECTORS.7 End of topic test End of topic test Go online Q4: If d = 4, e = and f =. 8 What is d e + /f in unit vector form? Q: The vector a has components 4. What are the components of a unit vector parallel to a? Q: Given the points A(,,), B(,-,) and C(-, 4, 7 ). a) Choose the conditions for collinearity of the points ABC. B is a common point. B is the midpoint of AC. There is a common point between the two vectors chosen. Two vectors are parallel. b) Find the component form of AB and state its z component. c) Find the component form of BC and state its z component. d) Are the points collinear? Q7: Find the position vector of the point P which divides the line AB in the ratio : and where A = (, -, 4 ) and B = ( -4, 4, - ). Q8: Given the position vectors to A(-,-7,9) and to B(-,-8,). What is the scalar product a b? Q9: Given A(,,-), B(9,,) and C(-,4,-4). a) What is the scalar product of the vectors AB and AC?
TOPIC. VECTORS 4 b) Find the angle BAC. Q: The points A(,,), B(4,-,4) and C(,-4,) are collinear. In which ratio does B divide AC? Q: OABCDEFG is a cube with side 4 units. G has coordinates (,4,4). U is the centre of face ODGC. V is the centre of face CGFB. a) What are the coordinates of B? b) What are the components of position vector u of the point U? c) What are the components of position vector v of the point V? d) What is the size of angle UOV? Q: A is the point (7,,7), B(, -,)and C(8,-,z). ABC is a right angle. What is the value of z? ABCDE is a square pyramid with the base sides of length 9 units. The sloping faces are all equilateral triangles. Q: If the vector BA = s, the vector BE = r and the vector BC = t, evaluate s t.
4 TOPIC. VECTORS Q4: Evaluate s (t + r). Q: Express the vector AD in terms of r, s and t. Q: Hence deduce the value of angle DAB, in degrees.
GLOSSARY 47 Glossary collinearity when a set of points lie on a single straight line scalar a scalar is a number that is used to measure size or how big or small something is scalar product in component form ( dimensions) ( ) a if p = and q = then the scalar product is the number p q =ad+be b scalar product in component form ( dimensions) a d if p = b and q = e then the scalar product is the number p q =ad+be c f +cf section formula the section formula states that if p is the position vector of the point P which divides n AB in the ratio m : n then p = m + n a + m m + n b
48 ANSWERS: TOPIC Answers to questions and activities Vectors Vector journeys exercise (page 7) Q: a) a b) b c) -a d) -b e) a-b f) b-a Q: a) e b) e c) f+e d) e-f Q: a) x b) /4w c) /4w x d) Steps: TQ = TS + SR + RQ What is TS? What is SR? What is RQ? Use these answers to find TQ. Answer: x e) Steps: What is RS? RS = 4RP change the subject of the formula to RP. What is RP? SP = RP RS Use these answers to find SP. Answer: x f) Steps: PT = PS + ST Use your answers to b) and e) to find PT. Answer: /4w x
ANSWERS: TOPIC 49 g) Steps: PQ = PT + TQ Use your answers to c) and f) to find PQ. Answer: w 4x Answers from page. Q4: e = ( 4 7 ) means go along 4 then down 7. Add, subtract and multiply vectors in D and D exercise (page ) Q: ( ) a) a + b = 4 ( ) b) a - b = 4 ( ) 9 c) a - c = 4 ( 4 d) 4a + b -c = 7 ) Q:. a + b = 4. b - c =
ANSWERS: TOPIC. 4c = 7 4. a - b = 7 8 7. a -b + c = Magnitude of vectors exercise (page ) Q7: a) b) c) 7 d) Steps: ( ) What are the components of a + b? Now find the magnitude of this resultant vector. Answer: 8 e) Steps: ( ) What are the components of b - c? Now find the magnitude of this resultant vector. Answer: 7 f) Steps: ( ) What are the components of a -b + c? 4 Now find the magnitude of this resultant vector. Answer: 4 Q8: a) 7 b) 4 c) 4 d) Steps: What are the components of u + w?
ANSWERS: TOPIC Now find the magnitude of this resultant vector. Answer: e) Steps: What are the components of v - u? Now find the magnitude of this resultant vector. Answer: 4 f) Steps: What are the components of v - w + u? Now find the magnitude of this resultant vector. Answer: 7 Position vectors exercise (page ) Q9: a) b) c) d) ( ) 7 ( ) ( ) 4 8 ( 4) + 8 = 8 or 4 Q: a) b) 7 c) 4 d) ( 7) + 4 + = 9 =
ANSWERS: TOPIC Q: Q: ( - ( - )) + ( ) + ( ) = 4 Q: ( 8) + ( 7) + ( - ( - 4)) = Unit vectors exercise (page ) Q4: i +j 7k Q: i +j +9k Q: a) Steps: What is b? Hint: magnitude of b. ( - ) + + 8 = What is a? If a and b are parallel if a = kb, what is the scalar k? = reciprocal of Use the scalar to find the components of a. Answer: b) Hint: Q7: Hint: 4 change components to i, j, k form. Answer: - i + 4 k Write down an expression for and make it equal to and solve for y. ( ) ( + y 7 + 7) = ( ) ( + y + ) = 7 7 9 49 + y + 49 = y = 4 49 y = 4 49 Answer: ± 7
ANSWERS: TOPIC Zero vectors practice (page 4) Q8: + 4 4 + 4 = = + 4 4 + 8 = Parallel vectors practice (page ) Q9: If we factorise t we get,t = which gives t = s vectors s and t are parallel. Q: v = 4 = w = p 8 4 = Since v and w are parallel they are both scalars of the same vector. Hence = / p and p =. p 4 Parallel vectors exercise (page ) Q: Yes, each component of the first vector is 4 times the second vector. Q: Steps: Since parallel vectors are multiples of each other ie. a =kbfortwovectors a and b, examine the y-component given and the value of b in the vector what is the value of k. k = 8 Answer: i j 4k Q: a) b) c) 49 Q4: Yes. Each component of the first vector is times the second vector. Q: -i j 4k
4 ANSWERS: TOPIC Collinearity practice (page 7) Q: PQ = q p = = 7 QR = r q = 7 8 = 4 = QR = PQ so QR and QR are parallel but Q is a common point P, Q and R are collinear. Q7: FG = g f = 8 4 = = EF = f e = a 7 4 = a Since E, F and G are collinear and FG = EF, it follows that a = and a =. Collinearity exercise (page 7) Q8: There is a common point between the two vectors chosen; Two vectors are parallel. Yes. The vector AB is times the vector BC and B is a common point here.
ANSWERS: TOPIC Q9: Steps: What are the components of PQ? PQ = What are the components of QR? QR = How many times PQ is QR? 7 What is x-equal to? Make x the subject to find its value. Answer: 8 = 8 9 x 8 = 9 x = 7 x Division of vectors in a given ratio exercise (page ) Q: a) Steps: The vector AP = m / n times the vector AB. State the fraction m / n. 4 Find AB and give the x-component. -4 Answer: - b) - c) -4 Q: a) Steps: The vector AT = m / n times the vector AB. State the fraction m / n. Find AB and give the x-component. Answer: -4 b) -7 c) - Q: a) Steps: The vector AT = m / n times the vector AB. State the fraction m / n. 74 Find AB and give the x-component. Answer: 4 b) 8
ANSWERS: TOPIC c) 48 Q: Steps: What is UV? 7 What is VW? 9 4 7 Use your answers to find the ratio. = = Answer: : or : Q4: Steps: Cross multiply, substitute position vectors and make g the subject of the formula. FG = GH g f = h g What is g equal to? g = f + h g = (f + h) Use this answer and substitute the components to find the coordinates of G. + Answer:(,,4) + 8 + The scalar product: Component form practice (page ) - a =, b = - and - Q: - c = - - 4 b + c = a (b + c) = ( - 4) + + ( - ) = - 8 + + ( - ) = -
ANSWERS: TOPIC 7 a b = ( - ) + ( - ) + ( - ) = - + + ( - ) = - 9 a c = ( - ) + + ( - ) ( - ) = - + + = - a b + a c = - 9 + ( - ) = - Hence a (b + c) = a b + a c a b Q: a b = = a b + a b + a b a b and b a = b b a a a b = b a + b a + b a b a But, a b +a b +a b =b a +b a +b a by the laws of algebra. Thus a b = b a The scalar product: Component form exercise (page ) Q7: a) - b) 94 c) Q8: Steps: a If p = b and q = c product a b? ad+be+cf d e, what is the correct formula for the scalar f Answer: Q9: Steps: If p = ai + bj + ck and q = di + ej + fk, what is the correct formula for the scalar product a b? ad+be+cf Answer: -
8 ANSWERS: TOPIC Q4: Steps: What are the components of a? a = 7 PQ = q p = What are the components of b? b = PR = r p = Use your answers to calculate a b. 4 4 = = Answer: 4 ( ) + + = 4 The scalar product: Geometric form exercise (page 8) Q4: Steps: What is the correct formula for the scalar product a b, where t is the angle between the vectors. a b cos (t) Answer: 88 Q4: Steps: What is the correct formula for the scalar product c d, where t is the angle between the vectors. c d cos (t) Answer: 84 Q4: a) Steps: a If p = b and q = c product a b? ad+be+cf Answer: -7 b) Steps: What is the value of a? 7 What is the value of b? 9 d e f, what is the correct formula for the scalar If θ is the angle between the vectors, calculate cos θ using the correct formula and leave as a fraction. -7 /
ANSWERS: TOPIC 9 Q44: Steps: Answer: 4 Find the vector from A to B and give it using standard basis. i +4j k Find the vector from A to C and give it using standard basis. i +j +k Now use the formula for cos θ where θ is the angle between the two vectors just calculated. Answer: 7 4 Q4: Steps: What is the magnitude of e? What is the magnitude of f? What is the angle between e and f? What is the exact value of cos? / Answer: cos = 9 / Q4: (,,4) Q47: (,,) Q48: = 4 4-4 Q49: Steps: What is GB GD? 4 + ( ) + ( 4) ( 4) = Answer: 4 = - - 4 Q: Steps: What is? GB 4 + + ( - 4) = What is GD? + ( - ) + ( - 4) = Substitute your answers into the formula for the scalar product to find the size of the angle BGD. ( ) Answer: cos =.
ANSWERS: TOPIC Perpendicular vectors exercise (page 4) Q: Steps: Which equation is used to show that the two vectors a and b are perpendicular? a b = Answer: - Q: Steps: Which equation is used to show that the two vectors a and b are perpendicular? a b = Answer: Q: Steps: Which equation is used to show that the two vectors a and b are perpendicular? a b = Answer: -9 End of topic test (page 44) Q4: j - k Q:. What is the magnitude of a? ( ) a = + 4 + ( - ) = + + 4 = =. What fraction of a is parallel unit vector? / a = Q: 4 a) There is a common point between the two vectors chosen. Two vectors are parallel. b) - c) 4 d) Yes. The vector AB = 8BC and B is a common point here.
ANSWERS: TOPIC Q7: Steps: The vector AP = m n times the vector AB state this fraction. / Answer: P(-,, ) Q8: Steps: a If p = b and q = c product a b? ad+be+cf d e, what is the correct formula for the scalar f Answer: a b =84 Q9: a) Steps: Find the vector from A to B and give it using standard basis. i + j + 7k Find the vector from A to C and give it using standard basis. i +j +k Answer: a b = - b) Steps: Find the vector AB giving your answer in unit vector form. 9.94 Find the vector AC giving your answer in unit vector form..4 Now use the formula for cosθ where θ is the angle between the two vectors just calculated. Answer: BAC =.88 Q: Steps: What are the components of AB? What are the components of BC? Factorise AB and BC to find the ratio. What is the common factor of AB? What is the common factor of BC? 4 - = 4 4-4 ( - ) 4 4 - - - - = - - - 4 -
ANSWERS: TOPIC Answer: B divides AC in the ratio :. Q: a) (4,4,) b) c) 4 d) Steps: What is u v? What is u? What is v? ( Answer: cos ) 8 4 = Q: Steps: 7 4 What are the components of BA? ( - ) = 7 8 What are the components of BC? - ( - ) = - 4 z z What is an expression for BA BC? 4 4+ (-4) + (z - ) = z - What is for perpendicular vectors? Answer: Q: Steps: a b Use the general formula cos θ = a b rearranged to give a b Answer: 4 Q4: 8
ANSWERS: TOPIC Q: Steps: In triangle ADC what, in terms of the other sides, is vector AD? AC + CD In triangle ACB, find AC in terms of the other sides and state AC in terms of a combination of r, s and t. -s + t In terms of the vectors r, s and t give a value for the vector CD. r Answer: r + t - s Q: Steps: In terms of r, s and t find AD AB. Evaluate this scalar product. Answer: 9 AD AB = ( - s + r + t) ( - s) = - s - s + - s r + - s t = 8 4. 4. =