Tutorial 1 Vector Calculus FDM1033, May 2015

1. Find the magnitude of the vector PQ with )2,1(P and )5,5(Q .

2. Find the length of the vector 7,3,2 v

.

3. Given the points in 3-dimensional space, )2,3,1(),7,5,3(),5,1,2( RQP and )0,1,2(S . Does

RSPQ ?

4. Find a vector of magnitude 5 in the direction of kjiv 253

.

5. Given kjiu 63

and kiv 12

, find

(a) vu

,

(b) the angle between vectors u

and v

,

(c) the vector vproju

,

(d) the scalar component of v

in the direction of u

.

6. Given )1,0,2(),3,1,1( QP and )1,2,0( R , find

(a) the area of the triangle determined by the points P, Q and R.

(b) the unit vector perpendicular to the plane PQR.

7. Find the volume of the parallelepiped determined by the vectors 3,2,2,0,1,4 vu

and

5,2,0w

.

8. Find the area of the parallelogram whose vertices are given by the points A (0, 0, 0), B (3, 2, 4),

C (5, 1, 4) and D (2, -1, 0).

9. Find the volume of the parallelepiped with adjacent edges AB, AC and AD with vertices A(-2, 1,

0), B(2, 3, 2), C(1, 4, -1) and D(3, 6, 1).

10. Find the parametric equation for the line of intersection of the planes 33 zyx and

1 zyx .

11. Find an equation of the plane that passes through (1, 2, 3) and contains the line tx 3 ,

ty 1 , tz 2 .

12. Find the equation of the line through (2, 1, 0) and perpendicular to both ji and kj .

13. Find the parametric equation of the line through the point (1, 0, 6) and perpendicular to

the plane x+3y+z=5.

14. Determine whether the given lines are skew, parallel or intersecting. If the lines are intersecting,

what is the angle between them?

3

3

1

6

1

2:2

1

2

2

3

2

1:1

zyxL

zyxL

15. Find the point in which the line x = 1 t, y = 3t, z = 1 + t meets the plane 2x y + 3z = 6.

16. Find the distance from the point Q (2, -3, 4) to the plane x + 2y + 2z = 13.

17. Find an equation of the plane containing the point (0, -2, -1) and parallel to the plane

342 yx .

18. Find an equation of the plane that passes through )3,2,1( and contains the line

tztytx 2,1,3 .

19. Find the equation of the plane through the point )1,4,2( A and orthogonal to vectors

2,1,0 v

and 1,1,1w

.

20. Use the scalar triple product to determine whether the points P(1,0,1), Q(2,4,6), R(3,-1,2) and

S(6,2,8) lie in the same plane.

21. Find the distance between the parallel planes given below:

P1: 02 zyx

P2: 05336 zyx

22. What is the equation of the plane that contains the line tztytx 2,1,3 and is parallel

to the intersection of the planes 1 zy and 02 zyx .