Worksheet A VECTORS 1 G H I D E F A B C

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Bertina Higgins
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1 Worksheet A G H I D E F A B C The diagram shows three sets of equally-spaced parallel lines. Given that AC = p that AD = q, express the following vectors in terms of p q. a CA b AG c AB d DF e HE f AF g AH h DC i CG j IA k EC l IB A B O C In the quadrilateral shown, OA = u, AB = v OC = w. Find expressions in terms of u, v w for a OB b AC c CB A B D C E F H G The diagram shows a cuboid. Given that AB = p, AD = q AE = r, find expressions in terms of p, q r for a BC b AF c DE d AG e GB f BH 4 R S O T The diagram shows parallelogram ORST. Given that OR = a + b that OT = a b, a find expressions in terms of a b for i OS ii TR Given also that OA = a that OB = b, b copy the diagram show the positions of the points A B.

2 Worksheet A continued A C D O B The diagram shows triangle OAB in which OA = a OB = b. The points C D are the mid-points of OA AB respectively. a Find simplify expressions in terms of a b for i OC ii AB iii AD iv OD v CD b Explain what your expression for CD tells you about OB CD. 6 Given that vectors p q are not parallel, state whether or not each of the following pairs of vectors are parallel. a p p b (p + q) (p 4q) c (p q) (p q) d (p q) (4q p) e ( 4 p + q) (6p + 8q) f (q p) ( q p) 7 The points O, A, B C are such that OA = 4m, OB = 4m + n OC = m + n, where m n are non-parallel vectors. a Find an expression for BC in terms of m n. The point M is the mid-point of OC. b Show that AM is parallel to BC. 8 The points O, A, B C are such that OA = 6u 4v, OB = u v OC = v u, where u v are non-parallel vectors. The point M is the mid-point of OA the point N is the point on AB such that AN : NB = : a Find OM ON. b Prove that C, M N are collinear. 9 Given that vectors p q are not parallel, find the values of the constants a b such that a ap + q = p + bq b (p + aq) + (bp 4q) = 0 c 4aq p = bp q d (ap + bq) (aq 6p) = 0 0 A O C D B The diagram shows triangle OAB in which OA = a OB = b. The point C is the mid-point of OA the point D is the mid-point of BC. a Find an expression for OD in terms of a b. b Show that if the point E lies on AB then OE can be written in the form a + k(b a), where k is a constant. Given also that OD produced meets AB at E, c find OE, d show that AE : EB = :

3 Worksheet B The points A, B C have coordinates (6, ), (, ) (, ) respectively O is the origin. Find, in terms of i j, the vectors a OA b AB c BC d CA Given that p = i j q = 4i + j, find expressions in terms of i j for a 4p b q p c p + q d 4p q Given that p = q =, find a p b q c p + q d q p 4 Given that p = i + j q = i j, find, in degrees to decimal place, the angle made with the vector i by the vector a p b q c p + q d p q Find a unit vector in the direction a 4 b 7 c d 4 6 Find a vector a of magnitude 6 in the direction i + j, b of magnitude in the direction 6i 8j, c of magnitude in the direction i 4j. 7 Given that m = i j n = µi j, find the values of λ µ such that a m + n = i j b m n = i + 8j 8 Given that 6i + cj, where c is a positive constant, find the value of c such that a r is parallel to the vector i + j c r = 0 d r = 9 Given that p = i + j q = 4i j, a find the values of a b such that ap + bq = i + j, b find the value of c such that cp + q is parallel to the vector j, b r is parallel to the vector 9i 6j c find the value of d such that p + dq is parallel to the vector i j. 0 Relative to a fixed origin O, the points A B have position vectors Find a the vector AB, b AB, c the position vector of the mid-point of AB, d the position vector of the point C such that OABC is a parallelogram. 6 respectively.

4 Worksheet B continued Given the coordinates of the points A B, find the length AB in each case. a A (4, 0, 9), B (,, ) b A (,, ), B (7,, ) Find the magnitude of each vector. Find a 4i + j 4k b i + j + k c 8i j + 4k d i j + k a a unit vector in the direction i j + 4k, b a vector of magnitude 0 in the direction i + j 0k, c a vector of magnitude 0 in the direction i 4j + k. 4 Given that λi + j 4k, find the two possible values of λ such that r = 4. Given that p =, q =, find as column vectors, a p + q b p r c p + q + r d p q + r 6 Given that i j k, find the values of λ µ such that a r is parallel to 4i + j 8k b r is parallel to i + 0j 0k 7 Given that p = i j + 4k, q = i + j + k i 4j 7k, a find p q, b find the value of k such that p + kq is parallel to r. 8 Relative to a fixed origin O, the points A, B C have position vectors ( i + 7j + 4k), (i + j + 8k) (6i j) respectively. a Find the position vector of the mid-point of AB. b Find the position vector of the point D on AC such that AD : DC = : 9 Given that λi λj k, that r is parallel to the vector (i 4j k), a show that λ = 0. Given also that r = 9 that µ > 0, b find the values of λ µ. 0 Relative to a fixed origin O, the points A, B C have position vectors respectively. a Find the position vector of the point M, the mid-point of BC. b Show that O, A M are collinear., 7 8 The position vector of a model aircraft at time t seconds is (9 t)i + ( + t)j + ( t)k, relative to a fixed origin O. One unit on each coordinate axis represents metre. a Find an expression for d in terms of t, where d metres is the distance of the aircraft from O. b Find the value of t when the aircraft is closest to O hence, the least distance of the aircraft from O.

5 Worksheet C Sketch each line on a separate diagram given its vector equation. a i + sj b s(i + j) c i + 4j + s(i + j) d j + s(i j) e i + j + s(i j) f (s + )i + (s )j Write down a vector equation of the straight line a parallel to the vector (i j) which passes through the point with position vector ( i + j), b parallel to the x-axis which passes through the point with coordinates (0, 4), c parallel to the line i + t(i + j) which passes through the point with coordinates (, ). Find a vector equation of the straight line which passes through the points with position vectors a 0 b 4 c 4 Find the value of the constant c such that line with vector equation i j (ci + j) a passes through the point (0, ), b is parallel to the line i + 4j (6i + j). Find a vector equation for each line given its cartesian equation. a x = b y = x c y = x + d y = 4 x e y = x f x 4y + 8 = 0 6 A line has the vector equation i + j (i + j). a Write down parametric equations for the line. b Hence find the cartesian equation of the line in the form ax + by + c = 0, where a, b c are integers. 7 Find a cartesian equation for each line in the form ax + by + c = 0, where a, b c are integers. a i (i + j) b i + 4j (i + j) c j (4i j) d i + j (i + j) e i j ( i + 4j) f (λ + )i + ( λ )j 8 For each pair of lines, determine with reasons whether they are identical, parallel but not identical or not parallel. a + s b + s 4 c + s 4 + t t t 9 Find the position vector of the point of intersection of each pair of lines. a i + j i b 4i + j ( i + j) c j (i j) i + j (i + j) i j (i j) i + 0j ( i + j) d i + j (i + 6j) e i + j ( i + 4j) f i + j (i + j) i j ( i + j) i 7j (i + j) i + j (i + 4j)

6 Worksheet C continued 0 Write down a vector equation of the straight line a parallel to the vector (i + j k) which passes through the point with position vector (4i + k), b perpendicular to the xy-plane which passes through the point with coordinates (,, 0), c parallel to the line i j + t(i j + k) which passes through the point with coordinates (, 4, ). The points A B have position vectors (i + j k) (6i j + k) respectively. a Find AB in terms of i, j k. b Write down a vector equation of the straight line l which passes through A B. c Show that l passes through the point with coordinates (, 9, 8). Find a vector equation of the straight line which passes through the points with position vectors a (i + j + 4k) (i + 4j + 6k) b (i k) (i + j + k) c 0 (6i j + k) d ( i j + k) (4i 7j + k) Find the value of the constants a b such that line i j + k (i + aj + bk) a passes through the point (9,, 8), b is parallel to the line 4j k (8i 4j + k). 4 Find cartesian equations for each of the following lines. a 0 b 6 c Find a vector equation for each line given its cartesian equations. a x = y + 4 = z b x y = 4 = z + 7 c x + = y + = z 6 Show that the lines with vector equations 4i + k + s(i j + k) 7i + j k + t( i + j + k) intersect, find the coordinates of their point of intersection. 7 Show that the lines with vector equations i j + 4k (i + j + k) i + 4j + k (i j + k) are skew. 8 For each pair of lines, find the position vector of their point of intersection or, if they do not intersect, state whether they are parallel or skew. a 0 b 0 6 c 8 8 d e f 0 7 8

7 Worksheet D Calculate a (i + j).(i + j) b (4i j).(i + j) c (i j).( i j) Show that the vectors (i + 4j) (8i j) are perpendicular. Find in each case the value of the constant c for which the vectors u v are perpendicular. c a u =, v = b u =, v = c 4 Find, in degrees to decimal place, the angle between the vectors c c u =, v = a (4i j) (8i + 6j) b (7i + j) (i + 6j) c (4i + j) ( i + j) Relative to a fixed origin O, the points A, B C have position vectors (9i + j), (i j) (i j) respectively. Show that ABC = 4. 6 Calculate a (i + j + 4k).(i + j + k) b (6i j + k).(i j k) c ( i + k).(i + 4j k) d (i + j 8k).( i + j 4k) e (i 7j + k).(9i + 4j k) f (7i j).( j + 6k) 7 Given that p = i + j k, q = i + j k 6i j k, a find the value of p.q, b find the value of p.r, c verify that p.(q + r) = p.q + p.r 8 Simplify a p.(q + r) + p.(q r) b p.(q + r) + q.(r p) 9 Show that the vectors (i j + k) (i + j 6k) are perpendicular. 0 Relative to a fixed origin O, the points A, B C have position vectors (i + 4j 6k), (i + j k) (8i + j + k) respectively. Show that ABC = 90. Find in each case the value or values of the constant c for which the vectors u v are perpendicular. a u = (i + j + k), v = (ci j + k) b u = ( i + j + k), v = (ci j + ck) c u = (ci j + 8k), v = (ci + cj k) d u = (ci + j + ck), v = (i 4j + ck) Find the exact value of the cosine of the angle between the vectors a 8 b 6 c 7 d 4 Find, in degrees to decimal place, the angle between the vectors a (i 4k) (7i 4j + 4k) b (i 6j + k) (i j k) c (6i j 9k) (i + j + 4k) d (i + j k) ( i 4j + k)

8 Worksheet D continued 4 The points A (7,, ), B (, 6, ) C (,, ) are the vertices of a triangle. a Find BA BC in terms of i, j k. b Show that ABC = 8. to decimal place. c Find the area of triangle ABC to significant figures. Relative to a fixed origin, the points A, B C have position vectors (i j k), (4i + j k) (i j) respectively. a Find the exact value of the cosine of angle BAC. b Hence show that the area of triangle ABC is. 6 Find, in degrees to decimal place, the acute angle between each pair of lines. a c b d Relative to a fixed origin, the points A B have position vectors (i + 8j k) (6i + j + k) respectively. a Find a vector equation of the straight line l which passes through A B. The line l has the equation 4i j + k ( i + j k). 8 b Show that lines l l intersect find the position vector of their point of intersection. c Find, in degrees, the acute angle between lines l l. 8 Find, in degrees to decimal place, the acute angle between the lines with cartesian equations x y z + = = 6 x 4 9 The line l has the equation 7i k (i j + k) the line m has the equation i + 7j 6k (i 4j k). = y + 7 = z a Find the coordinates of the point A where lines l m intersect. b Find, in degrees, the acute angle between lines l m. The point B has coordinates (,, ). c Show that B lies on the line l. d Find the distance of B from m. 0 Relative to a fixed origin O, the points A B have position vectors (9i + 6j) (i + j + k) respectively. a Show that for all values of λ, the point C with position vector (9 + λ)i + (6 λ)j k lies on the straight line l which passes through A B. b Find the value of λ for which OC is perpendicular to l. c Hence, find the position vector of the foot of the perpendicular from O to l. Find the coordinates of the point on each line which is closest to the origin. a i + j + 7k (i + j 4k) b 7i + j 9k (6i 9j + k).

9 Worksheet E Relative to a fixed origin, the line l has vector equation where λ is a scalar parameter. i 4j + pk (i + qj k), Given that l passes through the point with position vector (7i j k), a find the values of the constants p q, () b find, in degrees, the acute angle l makes with the line with equation i + 4j k (i + j k). (4) The points A B have position vectors respectively, relative to a fixed origin. a Find, in vector form, an equation of the line l which passes through A B. () The line m has equation + t Given that lines l m intersect at the point C,. b find the position vector of C, () c show that C is the mid-point of AB. () Relative to a fixed origin, the points P Q have position vectors (i j + k) (i + j) respectively. a Find, in vector form, an equation of the line L which passes through P Q. () The line L has equation 4i + 6j k (i j + k). b Show that lines L L intersect find the position vector of their point of intersection. c Find, in degrees to decimal place, the acute angle between lines L L. (4) 4 Relative to a fixed origin, the lines l l have vector equations as follows: l : l : i + k (i j + k), 7i j + 7k ( i + j k), where λ µ are scalar parameters. a Show that lines l l intersect find the position vector of their point of intersection. The points A C lie on l the points B D lie on l. Given that ABCD is a parallelogram that A has position vector (9i j + k), b find the position vector of C. () Given also that the area of parallelogram ABCD is 4, c find the distance of the point B from the line l. (4) (6) (6)

10 Worksheet E continued Relative to a fixed origin, the points A B have position vectors (4i + j 4k) (i j + k) respectively. a Find, in vector form, an equation of the line l which passes through A B. () The line l passes through the point C with position vector (4i 7j k) is parallel to the vector (6j k). b Write down, in vector form, an equation of the line l. () c Show that A lies on l. () d Find, in degrees, the acute angle between lines l l. (4) 6 The points A B have position vectors fixed origin O. 0 8 respectively, relative to a a Find, in vector form, an equation of the line l which passes through A B. () The line l intersects the y-axis at the point C. b Find the coordinates of C. () The point D on the line l is such that OD is perpendicular to l. c Find the coordinates of D. () d Find the area of triangle OCD, giving your answer in the form k. () 7 Relative to a fixed origin, the line l has the equation 6 + s 0 4. a Show that the point P with coordinates (, 6, ) lies on l. () The line l has the equation + t intersects l at the point Q., b Find the position vector of Q. () The point R lies on l such that PQ = QR. c Find the two possible position vectors of the point R. () 8 Relative to a fixed origin, the points A B have position vectors (4i + j + 6k) (4i + 6j + k) respectively. a Find, in vector form, an equation of the line l which passes through A B. () The line l has equation i + j k (i + j k). b Show that l l intersect find the position vector of their point of intersection. (4) c Find the acute angle between lines l l. () d Show that the point on l closest to A has position vector ( i + j k). ()

11 Worksheet F The points A B have position vectors 0 respectively, relative to a fixed origin. a Find, in vector form, an equation of the line l which passes through A B. () The line m has equation where a is a constant. a, Given that lines l m intersect, b find the value of a the coordinates of the point where l m intersect. (6) Relative to a fixed origin, the points A, B C have position vectors (i + j k), (i + j 7k) (6i + 4j + k) respectively. a Show that cos ( ABC) =. () The point M is the mid-point of AC. b Find the position vector of M. () c Show that BM is perpendicular to AC. () d Find the size of angle ACB in degrees. () Relative to a fixed origin O, the points A B have position vectors 9 7 respectively. a Find, in vector form, an equation of the line L which passes through A B. () The point C lies on L such that OC is perpendicular to L. b Find the position vector of C. () c Find, to significant figures, the area of triangle OAC. () d Find the exact ratio of the area of triangle OAB to the area of triangle OAC. () 4 Relative to a fixed origin O, the points A B have position vectors (7i j k) (4i j + k) respectively. a Find cos ( AOB), giving your answer in the form k 6, where k is an exact fraction. (4) b Show that AB is perpendicular to OB. () The point C is such that OC = OB. c Show that AC is perpendicular to OA. () d Find the size of ACO in degrees to decimal place. ()

the coordinates of C (3) Find the size of the angle ACB. Give your answer in degrees to 2 decimal places. (4)

the coordinates of C (3) Find the size of the angle ACB. Give your answer in degrees to 2 decimal places. (4) . The line l has equation, 2 4 3 2 + = λ r where λ is a scalar parameter. The line l 2 has equation, 2 0 5 3 9 0 + = µ r where μ is a scalar parameter. Given that l and l 2 meet at the point C, find the