1. Vector
Practice Questions

2. Question 1
Determine the magnitude and direction of the sum of two vectors u and v, if their magnitudes are 5 and 8 units respectively, and the angle between them is 30o

3. Question 2
Cassiopia is towing her friend on a toboggan using a rope which makes an angle of 25o with the ground. If Cassiopia is pulling with a force of 70N, what horizontal force is she exerting on the toboggan?

4. Question 3
Alice and Bob are towing their friends on a toboggan. Each is exerting a horizontal force of 60N. Since they are walking side by side, the ropes pull one to each side; they each make an angle of 20o with the line of motion. Determine the force pulling the toboggan forward.

5. Question 4

6. Question
Determine a unit vector for u=(1,2,3)

7. Question 5
Determine the angle between a=(1, -4) and b=(2, 5).

8. Question
Find a vector that is perpendicular to both a=(1, -4,5) and b=(2, 5, -3).
How about a unit vector?

9. Question Find the direction cosines and direction angles of the vector
The magnitude of u
Because vector is not used to point in the direction of the line, we have no requirement that the ‘z’ term be positive.

10. Question
Determine the angle, to the nearest degree, that (–4,1,5) makes with the positive x–axis.

11. Question

12. Question
Determine the coordinates of the point that divides AB in the ratio 4:2 where A is (2, -1, 4) and B is (3, 1, 7)
If AP:PB = m:n then

13. Understanding
Consider collinear points P, Q, and R. consider also a reference point O. Write OQ as a linear combination of OP and OR if:
Q divides PR (externally) in the ratio -3:8
Since:
Then: 8 Q P R -3 5

14. Question
Multiply by unit vector in direction of v.

15. Question Find the point of intersection of: Write in parametric form:
Therefore:
Solve first two
Check with third

16. Question
Determine the vector equation of a line that is parallel to 2x – 5y + 12 = 0 and has the same x–intercept as the line (x,y) = (2,-3) + t(3,-6).
Direction vector :
Point :
Vector Equation:

17. Question
Determine the scalar equation of the plane passing through the point (2,3,4) and parallel to the lines (x,y,z)=(1,1,1)+s(1,2,3) and (x,y,z)=(-2,4,5)+t(6,-1,2).
Direction vector :
Normal :
Scalar Equation:

18. Question
Determine the distance from the point P(1, -1, 2) to the plane 3x-2y+z=4

19. Question Determine the distance from the point P(2, -3, 2) to the line
First we need point on line: (1, -1, 1)
Now vector between two points:

20. Question
Determine the distance between two parallel planes x-y+z-10=0 and x-y+z+4=0
Find a point on one of the planes:
Put y=z=0 into x-y+z+4=0, and we obtain x=-4. Therefore a point is P(-4,0,0)
We now just need to determine the distance from P(-4,0,0) to x-y+z-10=0

21. Question
Determine whether the following points are collinear: A(3,1,0), B(1,0,2), C(5, 2, -2)
Find direction vector AB and AC and see if they are parallel:
Therefore AB=-AC, thus AB and AC are parallel
Since they both share the common point A, then A, B, C are collinear

22. Question
Determine the scalar equation of the plane perpendicular to plane x-y+z=3 containing the line (x,y,z)=(2,1,1)+t(2,1,3)
The normal to the plane x-y+z=3 is a vector on the required plane : (1, -1, 1)
Another vector on the plane is the direction vector of the line : (2, 1, 3)
The normal of the required plane is then :
The equation of the plane is then: 4x+y-3z+D=0
Substitute in the point on the line (2, 1, 1):
Therefore:

23. Question Given a line : And a plane: 2x-y-z+10=0
Determine the value k such that the line cuts the plane: a) at one single point
b) At many points
c) At no point: a)
The direction vector of line is perpendicular to normal of plane, so the line and plane are parallel, therefore no single solution.
b) Sub (1,2, k) into the plane: 2(1)-(2)-(k)+10=0 => k=10
c) If k does not equal 10, there is no solution

24. Question
Determine the area of the parallelogram below, using the cross product. 6 10
400

25. Question Determine the point of intersection of the three planes:
The point of intersection is: (-4,-3,0)