Created by T. Madas VECTOR PRACTICE Part B Created by T. Madas
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1 VECTOR PRACTICE Part B
2 THE CROSS PRODUCT
3 Question 1 Find in each of the following cases a) a = 2i + 5j + k and b = 3i j b) a = i + 2j + k and b = 3i j k c) a = 3i j 2k and b = i + 3j + k d) a = 7i + j + 4k and b = i + 3j+ 2k e) a = 2i + 5j 4k and b = 3i + j 3k a b, where the vectors a and b are i + 3j 17k, i + 4j 7k, 5i 5j+ 10k, 10i 18j + 22k, 11i + 18j + 17k Question 2 Find a unit vector perpendicular to both a = 2i + 5j + k and b = 3i j k ( 4 i 5 j 17 k )
4 Question 3 The vectors a and b, are not parallel. Simplify fully ( 2 + ) ( 2 ) a b a b. 5b a = 5a b Question 4 The vectors a, b and c are not parallel. Simplify fully ( + ) a b c a i. a i ( b c)
5 Question 5 The following vectors are given a = 2i + 3j k b = i + 2j+ k c = j+ 3k a) Show that the three vectors are coplanar. b) Express a in terms of b and c. a = 2b c
6 Question 6 The following three vectors are given a = i + 3j+ 2k b = 2i + 3j + k c = i + 2j+ λk where λ is a scalar constant. a) If the three vectors given above are coplanar, find the value of λ. b) Express a in terms of b and c. λ = 1, a = 3c b
7 Question 7 The vectors a, b and c are such so that c a = i and b c = 2k. Express ( + ) ( ) a b a b c in terms of i and k. 2i + 4k
8 CROSS PRODUCT GEOMETRIC APPLICATIONS
9 Question 1 Find the area of the triangle with vertices at A( 1, 1,2 ), B( 1,2,1) and ( 2, 3,3) C Question 2 Find the area of the triangle with vertices at A ( 2,1,1), B ( 1,0,4 ) and ( 3, 1, 1) C
10 Question 3 A triangle has vertices at A( 2, 2,0), B ( 6,8,6) and ( 6,8,12) C. Find the area of the triangle ABC. 90
11 Question 4 A parallelepiped has vertices at the points A( 2,1, t ), B ( 3,3,2 ), ( 4,0,5) E ( 1, 2,7), where t is a scalar constant. D and H G E F D C A B a) Calculate AB AD, in terms of t. b) Find the value of AB AD i AE The volume of the parallelepiped is 22 cubic units. c) Determine the possible values of t. ( t) ( t ) 12 3 i + 1 j 5k, 11t 44, t = 2,6
12 Question 5 A triangular prism has vertices at the points A ( 3,3,3), B( 1,3, t ), ( 5,1,5 ) F ( 8,0,10), where t is a scalar constant. C and F C D E A B The face ABC is parallel to the face DEF and the lines AD, BE and CF are parallel to each other. a) Calculate AB AC, in terms of t. b) Find the value of AB AC i AD, in terms of t. The value of t is taken to be 6. c) Determine the volume of the prism for this value of t. d) Explain the geometrical significance if t = 1. ( t ) ( t ) 2 6 i j+ 4k, 4t + 4, V = 14 cubic units, A, B, C, D are coplanar, so no volume
13 Question 6 A tetrahedron has vertices at the points A( 3,6,4), B ( 0,11,0 ), ( 4,1,28 ) D( 7, k,24), where k is a scalar constant. C and a) Calculate the area of the triangle ABC. b) Find the volume of the tetrahedron ABCD, in terms of k. The volume of the tetrahedron is 150 cubic units. c) Determine the possible values of k. area = 75, volume = k, k = 3,15
14 Question 7 With respect to a fixed origin O the points A, B and C, have respective coordinates k,8,6, where k is a constant. ( 6,10,10 ), ( 11,14,13 ) and ( ) a) Given that all the three points lie on a plane which contains the origin, find the value of k. b) Given instead that OA, OB, OC are edges of a parallelepiped of volume 150 cubic units determine the possible values of k. k = 10, k = 5,25
15 LINES
16 Question 1 Find an equation of the straight line that passes through the point P ( 1,4,1) and is parallel to the vector 3i + 4j k. Give the answer in the form r a = b where a and b are constant vectors. ( ) r 3i + 4j k = 8i + 4j 8k Question 2 Find an equation of the straight line that passes through the points P ( 5,0,9) and Q ( 8, 4,10). Give the answer in the form r a = b where a and b are constant vectors. ( ) r 3i + 4j+ k = 36i + 22j + 20k
17 Question 3 A straight line has equation ( ) r = 4i + 2j+ 5k + λ i + 8j 3k, where λ is a scalar constant. Convert the above equation into Cartesian form. x 4 y 2 z 5 = = Question 4 A straight line has equation ( ) r = 2i 3j + λ i 2j+ 2k where λ is a scalar parameter. Convert the above equation into Cartesian form. x 2 y + 3 = = z 1 2 2
18 Question 5 Convert the equation of the straight line into a vector equation of the form λ is a scalar parameter. x 3 y = = z r = a + λb, where a and b are constant vectors and ( ) r = 3i 2j+ 5k + λ 2i + 3j 7k Question 6 A straight line has equation ( ) ( 7 5 ) r i + j k i + k = 0 Convert the above equation into the form vectors and λ is a scalar parameter. r = a + λb, where a and b are constant x 5 z + 3 =, y = 2 7 5
19 Question 7 A straight line has equation ( ) = ( ) r i j k i j k Convert the above equation into the form vectors and λ is a scalar parameter. r = a + λb, where a and b are constant ( ) λ ( ) r = 3i 2j+ 2k + λ 2i 4j+ 3k or r = i + 2j k + 2i 4j + 3k Question 8 If the point A( p, q,1) lies on the straight line with vector equation ( ) = ( ) r i j k i j k, find the value of each of the scalar constants p and q. p = q = 3
20 Question 9 The straight line L has equation ( ) ( ) r 3i 2j + 3k 2i 3k + 4k = 0. Use a method involving the cross product to show that the shortest distance of the point 2, 1, 3 from L is 3 units. ( ) proof
21 PLANES
22 Question 1 Find a Cartesian equation of the plane that passes through the point A( 6, 2,5), and its normal is in the direction 5i 2j + 3k. 5x 2y + 3z = 49 Question 2 Find a Cartesian equation of the plane that passes through the point A ( 5,1,2 ), and its normal is in the direction 2i 7j+ k. 2x 7y + z = 5
23 Question 3 Find a Cartesian equation of the plane that passes through the points A ( 5,2,2), B ( 1, 2,1) and ( 3, 2, 2) C. 2x + 11y 12z = 8 Question 4 Determine a Cartesian equation of the plane that contains the point A( 9, 1,0 ) and the straight line with vector equation ( ) r = 5i + 2j + 2k + λ i 3j+ 6k, where λ is a scalar parameter. 24x + 26y + 9z = 190
24 Question 5 Find a Cartesian equation of the plane that contains the parallel straight lines with vector equations r1 = 2i + j + 5k + λ ( i + j+ k ) and 2 = µ ( + + ) where λ and µ are scalar parameters. r i j k i j k,. z x = 3 Question 6 Find a Cartesian equation of the plane that contains the intersecting straight lines with vector equations r = i + j+ k + λ ( i + j+ k ) and = µ ( + + ) where λ and µ are scalar parameters. r2 6i 2j 4k 2i j k, x 3y + z = 4
25 Question 7 a) Find a set of parametric equations for the plane that passes through the points 2,4,1 6,0, 2 C 0,1,7. A ( ), B ( ) and ( ) b) Eliminate the parameters to obtain a Cartesian equation of the plane. ( x, y, z) ( 2 2λ 4 µ,4 3λ 4 µ,1 6λ 3µ ) = + +, 33x + 18y + 20z = 158
26 GEOMETRIC PROBLEMS (WITH PLANES AND LINES)
27 Question 1 Find the coordinates of the point of intersection of the plane with equation x + 2y + 3z = 4 and the straight line with equation where λ is a scalar parameter. ( ) r = i + j 5k + λ i + j + 2k, ( 1,3, 1) Question 2 Find the coordinates of the point of intersection of the plane with equation 3x + 2y 7z = 2 and the straight line with equation where λ is a scalar parameter. ( ) r = 9i + 2j+ 7k + λ i + 3j+ 2k, ( 5, 10, 1)
28 Question 3 Find the size of the acute angle formed by the planes with Cartesian equations 4x + 4y 7z = 13 and 7x 4y + 4z = Question 4 Find the size of the acute angle between the planes with Cartesian equations 4x + 5y + 3z = 82 and 2x + 5y + 6z =
29 Question 5 Find the size of the acute angle between the plane with equation 2x 2y + z = 12 and the straight line with equation where λ is a scalar parameter. ( ) r = 7i j+ 2k + λ 2i + 6j 3k, 31.6 Question 6 Find the size of the acute angle formed between the plane with Cartesian equation 2x 2y z = 2 and the straight line with vector equation where λ is a scalar parameter. ( ) r = 2i + j 5k + λ 3i + 4j 12k, 14.9
30 Question 7 Find the size of the acute angle between the plane with equation 3x 2y + z = 5 and the straight line with equation where λ is a scalar parameter. ( ) r = 3i 2j+ 3k + λ 2i 3j+ 4k, 52.6 Question 8 Find shortest distance of the origin O from the plane with equation 4x + 3y 5z =
31 Question 9 Find shortest distance of the origin O from the plane with equation x + 2y + 2z = Question 10 Find shortest distance from the point A( 1,3, 2) to the plane with Cartesian equation x + 3y 5z =
32 Question 11 Find shortest distance from the point P ( 3,1,3 ) to the plane with Cartesian equation x y + 2z = 2. 6 Question 12 Find shortest distance of the point P ( 1,2,9 ) from the plane with Cartesian equation x + 4y + 8z = 16. 7
33 Question 13 Find the distance between the parallel planes with Cartesian equations 2x + 6y + 3z = 70 and 2x + 6y + 3z = Question 14 The straight line with vector equation ( λ 5) ( 2 λ ) ( λ 2) r = + i + j + + k, where λ is a scalar parameter, is parallel to the plane with Cartesian equation x + 2y + z = 10. Find the distance between the plane and the straight line. 6 6
34 Question 15 Find the distance between the parallel planes with Cartesian equations 3x + 2y + z = 20 and 3x + 2y + z = Question 16 Find the distance between the parallel straight lines with vector equations r = i + j k + λ ( i + j+ k ) and = + + µ ( + + ) where λ and µ are a scalar parameters. r2 2i k 5i 4j 3k,
35 Question 17 Find the distance between the parallel straight lines with vector equations ( 2 ) ( ) r i + k i + j k = 0 and x 4 = y 8 = z Question 18 Find the shortest distance between the skew straight lines with vector equations r1 = i + λ ( j+ k ) and 2 = µ ( 2 ) where λ and µ are a scalar parameters. r i j k i j k, 2 2
36 Question 19 Find the shortest distance between the skew straight lines with vector equations r = i + λ ( i k ) and = µ ( + ) where λ and µ are a scalar parameters. r2 3i 3j k i 3j k, Question 20 Find the shortest distance between the skew straight lines with vector equations r = i j+ k + λ ( j + k ) and = µ ( + ) where λ and µ are a scalar parameters. r2 i 2j 3k i 2k, 5 14
37 Question 21 Find the intersection of the planes with Cartesian equations giving the answer in the form 2x 2y z = 2 and x 3y + z = 5, r = a + λb, where a and b are constant vectors and λ is a scalar parameter. ( ) r = i 2j + λ 5i + 3j+ 4k Question 22 Show that the planes with Cartesian equations 4x + 5y + 3z = 82 and 2x + 5y + 6z = 124 intersect along the straight line with equation where λ is a scalar parameter. ( λ 6) ( 20 2λ) ( 2λ 2) r = i + j+ + k, proof
38 Question 23 The planes Π 1 and Π 2 have Cartesian equations: Π 1 : x 2y + 2z = 0 Π 2 : 3x 2y z = 5 Show that the two planes intersect along the straight line with Cartesian equation x 4 y 3 z 1 = = proof
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