Definition: A vector is a directed line segment which represents a displacement from one point P to another point Q.

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1 THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH Algebra Section : - Introduction to Vectors. You may have already met the notion of a vector in physics. There you will have thought of it as an arrow, that was used to represent a force. Adding forces corresponded to adding arrows. In this topic we are going to look at vectors from a geometric view point, although we will include some examples based on simple ideas from physics. One of the most powerful developments in Mathematics came from the simple idea of the co-ordinate plane. Indeed -dimensional co-ordinate geometry was crucial in the development of the Calculus. The obvious question arises as to how we can generalise this to higher dimensions. Vectors give us a way of generalising co-ordinate geometry into higher dimensions in a very straight forward manner. Definition: A vector is a directed line segment which represents a displacement from one point P to another point Q. The word vector comes from the Latin veho (cf. vehicle), meaning to carry. We represent a vector either using the notation PQ or by using v. In the algebra notes (and in these notes), vectors are represented using bold letters, v. You should represent vectors by underlining the letter, viz v. This is important, because you will need to carefully distinguish between vectors, scalars (and later matrices). v Q w R PQ SR P S A vector has both direction and length (or magnitude). Two vectors are equal if they have the same direction and the same magnitude. Hence in the diagram v = w. We will denote the length of the vector v by v.

2 Two vectors are parallel if they have the same direction. Position Vectors: We choose a fixed point O, in whatever dimensional space we happen to be and call this the origin. The position vector of a point in any number of dimensions will be represented by a vector from the origin to that point. P O Hence the vector OP in the diagram is called the position vector of the point P. A position vector gives the position of a point in space, whereas a direction vector is simply a vector having direction and magnitude (length). Addition of vectors: To geometrically add two vectors there are two different methods (each important). If we think of a force vector, then, the obvious way to add two vectors is to put them tip to tail and join the tail of the first to the tip of the second, as in the diagram. v + w w w v To add the vectors v and w, we move w and then complete the triangle. This method of addition is known as the triangle law of addition. You can see from this that one could obtain the same vector by forming a parallelogram from the two vectors and taking the diagonal (often called the resultant) as the sum of the two vectors.

3 v + w v w This method is known as the parallelogram law. Subtraction of vectors is performed in a similar way: w v v w To check this makes sense, add the vectors that are tip to tail, v+(w v) = w as expected. Observe that the vector labelled w v is not a position vector. Thus if P and Q have position vectors v and w respectively, then PQ= w v. In general, PQ= OQ OP. w v Q P w v O

4 Example: Suppose ABCDEF is a regular hexagon with the vector p on the side AB and vector q on the side BC. Express the vectors on the sides: CD, DE, EF, FA and the diagonals AC, AD, AE in terms of p and q. CP: In a paralleogram ABCD, AB= a, AD= b, and M is the intersection of the diagonals. Express, in terms of a and b the vectors, MA, MB, MC, MD. The Triangle Inequality: Let us restrict ourselves, for the moment, to the plane. Since the sum of any two sides of a triangle must exceed the third side, we can write for any vectors u and v. u + v u + v v u + v u

5 Q: When do we have equality? CP: By writing u = u + v v, prove that u + v u v. Scalar Multiplication: We can multiply a vector by a scalar λ (generally just a real number). This has the geometric effect of stretching the vector if λ >, stretching and reversing its direction if λ <. λu u CP: a. Suppose that a and b are non-collinear (non-zero) vectors and that λa + µb =. Deduce that λ = µ =. b. Generalise this result to three non-coplanar (non-zero) vectors. Commutative and Associative Laws: The commutative law of vector addition states that a + b = b + a. Geometrically this is obvious: a + b = b + a b a a b The associative law of vector addition states that a + (b + c) = (a + b) + c. Again the following geometric proof (?) will suffice.

6 a + (b + c) (a + b) + c = a + b b c b + c Dependence: a Suppose we have a collection of vectors a,a,...a n. Suppose we start from the origin and move along the vector, λ a, then along λ a, then along λ a and so on until we move along λ n a n. Can we choose the values of the λ i s NOT ALL ZERO so that we arrive back at the origin? In general- NO. However, if we can, then we say that the vectors a,a,...,a n are linearly dependent. In other words: The vectors a,a,...,a n are linearly dependent if there exist real numbers λ, λ,..., λ n, NOT ALL ZERO such that λ a + λ a λ n a n =. In the diagram, the vectors a,b,c are linearly dependent: c b a Example: Suppose a,b are non-zero vectors. Show that p = a+b,q = b c,r = a b+c and s = b + c are linearly dependent.

7 Simple Applications to Physics: Ex: The center of mass of a system of particles is a specific point at which, for many purposes, the system s mass behaves as if it were concentrated. Suppose masses m, m and m are placed at the points A, B and C respectively, with position vectors a,b and c. Let M, with position vector m be the centre of mass. A B M C 7

8 Geometric Proofs: Ex: Prove (using vectors) that the line joining the midpoint of two sides of a triangle is parallel to the third side and half its length. Ex: Prove that the medians of a triangle meet at a point G which divides each median in the ratio :. 8

9 CP: Suppose P, Q, R are the midpoints of AB, BC, CA respectivley in ABC. Let O be any point inside the triangle. Use vectors to prove that OP + OQ + OR = OA + OB + OC. Is the result still true if O is outside the triangle? Co-ordinates: Thus far, much of what we have done works in any number of dimensions. We are now going to define n dimensional space and introduce a co-ordinate system in which to place our vectors. We take an n-tuple a a.. a n of real numbers and think of each a i as lying on an axis x i. In and dimensions, we identify these axes as the XY and XY Z axes respectively, which are mutually orthogonal. The set of all such n-tuples will be called R n. We say that the vector PQ in R n has co-ordinates a a... a n, if we must move a units along the x axis, a units along the x axis, and so on, when moving from P to Q. a a Hence an n-tuple.. in Rn can be interpreted as the position vector of a point P a n in R n. For example, in R, the point P in the diagram has position vector. 9

10 P = We can then define the addition of two vectors (algebraically) in R n by a b a + b a.. + b.. = a + b.. a n b n a n + b n and multiplication by a scalar λ to be λ a a.. a n = Multiplying a vector by a scalar λ merely stretches the vector (if λ > ) or shrinks it if λ <. If λ is negative then the vector reverses direction. (Note: The algebraic definition of addition agrees with the geometric definition.) v λa λa.. λa n v v

11 Note that we can now prove such rules asthe commutative lawalgebraically, viz: a b a + b b + a b a a a + b =.. + b.. = a + b.. = b + a.. = b.. + a.. = b +a. a n b n a n + b n b n + a n b n a n Parallel Vectors. Two vectors are defined to be parallel if one is a non-zero multiple of the other. That is, v is parallel to w if v = λw for some scalar λ. For example, is parallel to. Ex: Find the vectors PQ, and QP if P = 7 and Q =. ( ) ( ) ( ) ( ) Ex: Suppose that A = B =, C =, D = are the position vectors for four points A, B, C, D. Prove that the quadrilateral ABCD is a parallelogram.

12 CP: ABCD is ( a parallelogram ) ( ) with vertices ( ) A, B, C, D which have the following position vectors: A =, B =, C =. Find the three possible position vectors of D. (Hint: Keep it simple!) Basis Vectors: ( ) ( ) The standard basis vectors in R are the vectors and which are often denoted by i and j. Observe ( ) that every vector in R can be written in terms of these basis vectors. For example can be written as i j. In -dimensions, the basis vectors, i,j,k are,,. a Note that every vector in R can be expressed in terms of these basis vectors, viz: a a can be expressed as a i + a j + a k. In higher dimensions, we label the basis vectors as e,e,... and so on. Thus, in R, we have e =,e =,e =,e =. Once again, we can represent any vector in R n in terms of the standard basis vectors in R n. Distances and Lengths: ( ) a Given a vector x = in R b, we can use Pythagoras Theorem to compute the length of this vector as a + b. We use the notation x = a + b. In R, given a vector a x = b, we can see from the diagram that OP = a + b and then in OAP we c have OA = x = a + b + c.

13 A c O a X b P In higher dimensions, we can define the length of a vector by generalising this formula, i.e. a a Definition: A vector x =. in Rn has length x given by x = a + a + + a n. Ex: Find the lengths of a = a n and b =.

14 The distance between two points A and B in R n will be defined as the length of the vector AB, a b a b in other words, if A has position vector a = then the length of AB is. a n and B has position vector b =. b n, AB = b a = (b a ) + + (b n a n ). ( ) ( ) Ex: Find the distance between and and between and. The length function, (sometimes called a norm) has the following properties:. a.. a = if and only if a =.. λa = λ a, for λ R. A vector which has unit length is called a unit vector. Any vector can be made into a unit vector by dividing by its length. Ex: Find a unit vector parallel to the vector.

15 CP: The point P = x y z Prove that cos α + cos β + cos γ =. makes angles α, β, γ respectively with the X, Y and Z axes. Ex: Suppose A and B are points with position vectors a and b. Find a vector (in terms of a and b) which bisects the angle AOB, where O is the origin.

16 Equations of Lines: We seek to find the equation of a line in vector form. The vector equation of a line is a formula which gives the position vector x of every point on that line. This equation is sometimes referred to at the parametric vector form of the line. I will generally just say vector equation of the line. Suppose we have a line passing through the origin which contains a vector u in R. Every point on that line will have a position vector which is a multiple of u. Conversely, every multiple of u will correspond to the position vector of a point on that line. Hence the equation of the line can be written as x = λu where λ is any real number. x = λu u ( ) For example, if u were the vector then the equation of the line through u passing ( ) through the origin would be x = λ, λ R. Another way of denoting the set of all real multiples of a given vector is to call it the span of the vector. Thus we could write {λu : λ R} as span(u). This idea of span is extremely important.

17 ( Ex: In R what is the span of ) (? What is span )? The advantage of this definition of the equation of a line is that it easily generalises to any number of dimensions. For example, the equation of the line in R which passes through the origin and is parallel to the vector is simply x = λ. If the line does not pass through the origin, then we proceed as follows: To find the vector equation of a line we need to know two things:. The position vector a of a point A on the line. The direction of the line, i.e. a vector AB= b parallel to the line. A b B x λb X x = a + λb a Thus, the span of b will give a line through the origin parallel to b and adding a will shift the line to its proper position. Thus we can find the position vector x of any point X on the line by going from the origin along the vector a and then moving along the line, by adding some multiple of b until we reach X. Thus, the vector OX is given by OX= OA + AX and AX is some multiple of b. Hence the equation of the line is x = a + λb, λ R. 7

18 Ex: Find the vector equation of the line passing through the point P with position vector and parallel to the vector. Ex: Find the vector equation ofthe line passing through the two points P, Q with position vectors P = and Q =. CP: Let A, B, C be a triangle and X be the midpoint of BC. Let Y be the point on AC which divides AC in the ratio :. Let T be the point on AX that divides AX in the ratio :. Prove that T lies on the line through Y B. Ex: Find the vector equation of the line in -dimensions with cartesian equation y = x+. 8

19 Ex: Does the point lie on the line x = + λ? Configurations: In R two lines can meet at a point be parallel neither meet nor be be parallel. CP: Two non-parallel lines in space that never meet are called skew lines. Prove that the lines x = + λ and x = + µ are skew lines. Line Segments: The set S = x R : x = represents that line segment from + λ to., λ 9

20 CP: Decide whether or not the line segment l : joining line segment l : joining and. and meets the Cartesian Equations of the Line: Given the vector form of a line in -dimensions, we can write down the Cartesian equations (note the plural) as follows. For example, suppose the vector equation is x = + λ. Recall that x is simply an abbreviation for x x. Hence, equating co-ordinates, we can write x = + λ, x = λ, x = + λ. x Eliminating λ from these equations we have x + = x = x. These are called the cartesian equations of the line. Clearly this can be done for any such line and so the general form is x a α = x b β = x c, γ where (a, b, c) is a point on the line and (α, β, γ) is a direction vector of the line, provided that none of the numbers α, β, γ is zero. a α Hence in vector form this would be x = b + λ β. c γ The following example tells us what to do when one of the components of the direction vector is zero. Ex: Convert x = + µ into cartesian form.

21 Observe that two lines will be parallel if their direction vectors are parallel. Two direction vectors are parallel if and only if one is a nonzero multiple of the other. For example the lines x = +λ and x = 7 +λ are parallel since their direction vectors are and first. Observe also that the equations x = respectively and the second vector is simply a multiple of the +λ and x = +λ represent the same lines, since they are parallel and pass through the same point. Ex: Find the equation of the line passing through and parallel to x + = y = z +. Ex: Find the intersection (if possible) of the lines x = x = + µ. + λ and

22 CP: Prove that the angle bisectors of a triangle are concurrent. Equations of Planes: In -dimensions and higher, we can construct planes. Suppose we seek the vector equation of the plane passing through the origin parallel to two given non-parallel vectors a and b. Q X µb b x a λa P To reach any point X with position vector x on the plane we need to stretch the vector a to P and stretch the vector b to Q in such a way that OX= OP + OQ. Thus, x = λa+µb. Conversely, if we take the vector which results from adding a multiple of a and a multiple of b then this will be the position vector of a point on the plane. Ex: Find the vector equation of the plane passing through the origin parallel to the vectors and 7. The plane generated by two such vectors is called the span of the two vectors. So for example,

23 the span of and 7 is simply the set λ + µ 7 : λ, µ R. This set is also referred to as the set of all linear combinations of the two vectors. Thus, given any two vectors a,b, span{a,b} = {λa + µb : λ, µ R} and we say that x is a linear combination of a and b if x = λa + µb for some particular λ and µ. Ex: Describe the span of,. Repeat for span,. As with the equation of a line, to get the vector equation of a plane not through the origin, we simply shift the plane by adding any position vector of a point which lies on the plane. Thus to obtain the vector equation of a plane we need:. The position vector of a point on the plane.. Two non-parallel vectors which are parallel to (or lie on) the plane. Ex: Find the vector equation of the plane passing through the point P with position vector and parallel to the vectors and.

24 Ex: Find the vector equation of the planepassing through the three points P, Q, R with position vectors P =, Q = and R = 7. Configurations: In R, two (distinct) planes can be parallel meet in a line In R, three (distinct) planes can be arranged so that the three planes are parallel two planes are parallel and the third plane is parallel to neither the first two. they meet at a single point they meet in a line none are parallel, but no point lies on all three planes. In the next chapter we will learn how to analyse these scenarios algebraically.

25 Regions: Ex: The set S = x = λ + µ represents the parallelogram with vertices O, Ex: The set S = x = λ + µ represents the triangle with vertices O,, λ, µ,,, in R., λ, µ λ in R CP: Suppose that a,b,c are the position vectors of non-collinear points in R. Let S = {x R : x = αa + βb + γc, where α, β, γ and α + β + γ = }. Describe S in geometric terms and give a proof that your claim is correct. Cartesian Equation of a Plane: As with lines, we can find the cartesian equation of a plane by eliminating the two parameters λ and µ. This is generally quite fiddly to do algebraically. Later in this course, you will see a much better method, but for the moment, we will do it by algebra. Ex: Find the cartesian equation of the plane x = + λ + µ.

26 The procedure can be reversed to find the vector equation of a plane from the cartesian equation. Ex: Find the vector equation of the plane x y + z =. Further Examples: Ex: Find the intersection of the planes x + y z = and x + y + z = 9.

27 Ex: Show that the line x = λ x = + µ + ν. is parallel to the plane Ex: Find the intersection of x = + λ and x + y z = 9. 7