TUTORIAL VECTORS

CHAPTER 5 : VECTORS SM025

TUTORIAL 5.1 : VECTORS IN THREE DIMENSIONS
5.2 : SCALAR PRODUCT

LEARNING OUTCOMES :
5.1 a) State the types of vectors.

b) Find the magnitude of a vector and unit vector.
c) Perform addition, subtraction and scalar multiplication of vectors.
5.2 a) Find the scalar product.
b) Use the properties of scalar product.
c) Find the angle between two vectors.
d) Find the direction cosines and directions angles for a non-zero vector.

REFERENCE : Lecture 1

1. Given vector a = 2 i − 2 j + k , find
a) the magnitude of a
b) the unit vector of a .

2. Given vectors a = 2 i − 3 j + 5 k and b = 3i + 4 j − 2 k , find
a) a + b
b) 2a − 4b
c) a  b

3. Given three vectors a = 8i − 2 j + 5k , b = 2i +3 j +  k and c = −4i + j +  k . Find

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CHAPTER 5 : VECTORS SM025

a)  if a and b are perpendicular.
b)  if a and c are parallel.
4. Given a = 3 , b = 5 and a  b = 9 , find a − b .
5. If u , v and w are three non-zero vectors such that u + v + w = 0 , show that

w2− u 2− v 2
uv =

2
6. If u + v = 5 and u − v = 1, find u  v .
7. Find the angle between the vectors a = 2 i − 3 j + 5 k and b = 3i + 4 j − 2 k .
8. Find the direction cosines and direction angles of vector a = 2 i − j + 3 k . Hence, show

that cos2  + cos2  + cos2  = 1.

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CHAPTER 5 : VECTORS SM025

TUTORIAL 5.3 : VECTOR PRODUCT

LEARNING OUTCOMES :
a) Find the vector product.
b) Use the properties of vector product.
c) Find the area of a parallelogram and a triangle.

REFERENCE : Lecture 2

9. If a = 2i + 4 j − k and b = 3i − 2 j + 4k , find a  b .
10. Given vectors a = 2i + j + q k and b = qi − 2 j + 2q k . Find the value of q if

ab=8i− 4 j − 6k .
11. Find the unit vectors which is perpendicular to both vectors a = 2i + 3 j + 6 k and

b=i− j+ 2k.
12. If a + b + c = 0 , show that a b = b  c = c  a .
13. Given that a , b and r are three vectors and  is a scalar such that

a  r = b + a and a  r = 2 . Show that r = 2a − a  b .
a2

14. Show that a  b 2 = a 2 b 2 − ( a  b )2 .
15 a) Find the area of parallelogram with sides a = i + j − 4k and b = 5i − 4 j − 2k .

b) Find the area of triangle with vertices A(3, 5, 1) , B(6, 3, 3) and C(-1, 3, 5).

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CHAPTER 5 : VECTORS SM025

TUTORIAL 5.4 : APPLICATIONS OF VECTORS IN GEOMETRY

LEARNING OUTCOMES :
a) Find the equation of a straight line.
b) Determine the angle between two straight lines.

REFERENCE : Lecture 3

16. Find the vector, parametric and Cartesian equations of the line passes through point
A(1, − 4, 2) and parallel to 2i − j + 3k .

17. Find the vector equation of line passes through points A(1, 2, 1) and B(−1, 3, 2) .

18. A straight line with vector equation r = 2i − j + k + t(i − j + 2k) passes through point

(a, 1, b) . Find the values of a and b .

19. Find the angle between two straight lines r1 = (2 i + 2 j − 4k) + t(i + j + k) and
r2 = ( i + 2 j + k) + s(2i − 3 j + 4k) .

20. Find the angle between the straight lines l1 and l2 which are defined by

l1 : x = 1+ 2t , y = 3 − t , z = 5 − 2t and l2 : x−2= y +1 =1− z .
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CHAPTER 5 : VECTORS SM025

TUTORIAL 5.4 : APPLICATIONS OF VECTORS IN GEOMETRY
LEARNING OUTCOMES :

c) Find the equation of a plane.
REFERENCE : Lecture 4

21 a) Find the vector equation of the plane with normal vector 5i − 3 j + 7 k and
containing the point A(−5, 1, 2) .

b) Find the vector equation of the plane passes through point A(1, 2, − 3) and
perpendicular with line x = t , y = 2 − 2t , z = 1 + 3t .

22. Find the Cartesian equation of the plane passes through the points A(−1, 3, 1) ,
B(2, 1, 2) and C(4, 2, − 1) .

23. Find the Cartesian equation of the plane passes through the point A(4, 0, − 2) and
perpendicular to the planes x − y + z = 0 and 2x + y − 4z = −5.

24. Find the Cartesian equation of the plane
a) passes through point A(6, − 7, 4) and parallel to the xz − plane.
b) contains point A(2, 6, − 1) and parallel with the plane 4x − 2y + z = 1.

25. Find the Cartesian equation of the plane containing the line
r = 2i + 4 j − 3 k + t(2i − 3 j + 4k) and the point A(7, − 3, 5) .

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CHAPTER 5 : VECTORS SM025

TUTORIAL 5.4 : APPLICATIONS OF VECTORS IN GEOMETRY

LEARNING OUTCOMES :
d) Determine the angle between
i) two planes
ii) a line and a plane
e) Determine the point of intersection between a line and a plane.

REFERENCE : Lecture 5

26. Find the angle between two planes below
a) 2x − y + z = 5 and 3x + y − z = 1
b) r  (2i − 4 j + 6k) = 1 and r  (2i − j + k) = 7

27. Prove that the planes 2x − 3y − z = 5 and − 6x + 9y + 3z = −2 are parallel.
28. Find the angle between the line and the plane below

a) r = 2i + 3 j − k + t(2i − j − 2k) and 2x + 4y − z = 1
b) x = 2 , y − 1 = z − 2 and r  (i + 2 j − k) = 11

23
29. Find the intersection point between the line r = 2i + j − k + t(i + 2 j + k) and the plane

2x + y − 2z = 13.
30. Find the intersection point between the line x − 2 = y = z − 1 and the plane

1 2 −2
r  (4i + j + 8k) = 26 .

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CHAPTER 5 : VECTORS SM025

ANSWER b) 1 (2i − 2 j + k) c) − 16
1 a) 3 3
2 a) 5i + j + 3k
b) − 8i − 22 j + 18k

3 a)  = −2 b)  = − 5
2

4) 4

6) 6

7) 118.80

8) cos  = 2 , cos  = − 1 , cos  = 3 ;
14 14 14

 = 57.70 ,  = 105.50 ,  = 36.70

9) 14 i − 11j − 16 k

10) q = 2

^ 1 ^ 1 (12i + 2 j − 5k)

11) u = (12i + 2 j − 5k) or u = −
− 173 − 173

15 a) 27 unit 2 b) 12.4 unit 2

16) r = i − 4 j + 2 k + t(2i − j + 3k) ; x = 1 + 2t , y = −4 − t , z = 2 + 3t ;

x −1 = y + 4 = z − 2
2 −1 3

17) r = i + 2 j + k + t(−2i + j + k)

18) a = 0 , b = −3
19) 71.20
20) 85.80

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CHAPTER 5 : VECTORS SM025

21 a) r  (5i − 3 j + 7k) = −14 b) r  (i − 2 j + 3k) = −12

22) 5x + 11y + 7z = 35

23) x + 2y + z = 2

24 a) y = −7 b) 4x − 2y + z = −5

25) 4x + 4y + z = 21

26 a) 60.50 b) 40.20
28 a) 8.40 b) 6.50
29) (5, 7, 2)

30) (1, − 2, 3)

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