VECTORSExercises

1. Three points A, B and C have position vectors given respectively by

a = 7i + 4j – 2k, b = 5i + 3j – 3k, c = 6i + 5j – 4k, find the angle BAC

Ans: 60o

2. Find the area of the parallelogram whose adjacent sides are 5i – 2j + k and –2i + j + 2k .

Ans: 18.41

3. Find the area of the triangle having the vertices A(1, 1. 1), B (0, 1, 2) and C(3, 2, 1).

Ans: 1.225

4. Find a unit vector perpendicular to both of the vectors m = 2i – 3j + k, and n = 4i – 2j + k.

Ans:

5 Find the Cartesian equation of the line that is parallel to the vector –2i + 5j + 6k and passes through

the point A with position vector i – 3j + 2k.

Ans:

6. Find the vector equations of the lines with the following Cartesian equations:

(a) (b)

Ans: (a) r = 2i + j – k + (3i + j + 6k) (b) r = i + 2j + k + ( i + 5 j + k)

7. Find the vector and Cartesian equations of the straight line joining AB given that A(2, –1, 5) and

B(–1, 4, 3).

Ans: r =2i – j + 5k + (–3 i + 5 j – 2k), .

8. Find the vector and Cartesian equations of the plane that passes through P(–1, 3, 1) and is perpendicular

to the vector n = i – j – 3k.

Ans: r . (i – j – 3k) = –7, x – y – 3z = –7

9. Find the Cartesian equation of the plane with the vector r . (i – 3j – 5k) = 7.

Ans: x – 3y – 5z + 7

10. Find the vector equation of the plane containing the three points A, B and C whose position vectora are

2i + j – k, 3i + j + k and i – 2j + 3k respectively.

Ans: r . (–2i + 8j + 3k)= 1.

11. Find the vector and Cartesian equations of the plane that is perpendicular to 2i – 3j + k. and contains

the point with the position vector i – 3j – 2k.

Ans: r . (2i – 3j + k) = 9, 2x – 3y + z = 9

12. Find the angles between the following pairs of planes, L1 and L2.

(a) L1 : r = 2i + j + ( i + j) (b) L1 : r = 2i + j – 3k + (i – j + k)

L2 : r = –3i – j + (2 i –j) L2 : r = 2i + j – 3k + (– i + j – k)

Ans: (a) 71.6o (b) 0o

13. Find the acute angle between the two straight lines

L1 : and L2 : .

Ans: 80.4o.

14. Find the acute angle between the line and then plane 2x + 4y – z =1.

Ans: 61.0o

15. Find the acute angle between the two planes, 2x + 4y – 2z + 5 = 0 and r . (3i – j + k)= 4.

Ans: 90o

16. The lines L1 and L2 have Cartesian equations and

respectively. Show that L1 and L2 intersect and find the position vectorof the point of intersection.

Ans: (4, 5, 9)

17. Find the point of intersection of each of the following pairs of lines:

(a) L1 : r = –2j + 8k + ( i – j + k) (b) L1 : r = i + 3j + k + (i – j + k)

L2 : r = i + 3j + k + (2 i –j) L2 : r = i – j + 2k + (– i + 2j – k)

Ans: (a) (–3, 1, 5) (b) (3, –5, 4)

18. Find the point of intersection between the line and the plane 3x + 2y + z = 16.

Ans: (5, –3, 7)

19. Find the point of intersection between the line and the plane

3x – 2y + z = 5. [3]

Solution: the point of intersection is (4, 3, –1).

20. Find the Cartesian equation of the line of intersection of the two planes x + y – z = 3 and 3x + 4y = 1.

Ans: