Vector

Practice Questions

Question 1Determine 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

30

5u

8v

150

2 2 22 cos 150u v u v u v

2 25 8 2 5 8 cos 150

158.28

12.6u v

sin sin 150

5 12.65sin 150

s

11.4

in12.6

Question 2Cassiopia 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?

70cos 25h

63.4N

25h

Question 3Alice 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.

60N

60N

140

60cos 20 60cos 20r

112.8N

2020

Question 4

If and3 2 4 2 2 2 5 find u i j k l v i j k l u v

2 2 2 2

2 5 2 3 2 5 4 2 2

2 1,3, 1,2 5 4, 1,2, 2

2,6, 2,4 20, 5,10, 10

22,11, 12,14

22 11 12 14

945

3 105

u v i j k l i j k l

Question

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

2 2 2

ˆ

1,2,3

1 2 3

1,2,3

14

1 2 3, ,

14 14 14

uu

u

Question 5

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

1

1

2 2 22

cos

cos

cos

1 2 4 5cos

1 4 2 5

144.2

a b a b

a b

a b

a b

a b

Question

Find a vector that is perpendicular to both a=(1, -4,5) and b=(2, 5, -3).

1, 4,5 2,5, 3

13,13,13

v a b

How about a unit vector?

2 2 2

ˆ

13,13,13

13 13 13

13 13 13, ,

507 507 507

vv

v

Find the direction cosines and direction angles of the vector

The magnitude of u

Question

0,5, 3u

2 2 20 5 3

34

u

0cos

345

cos343

cos34

90

31

121

Because vector is not used to point in the direction of the line, we have no requirement that the ‘z’ term be positive.

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

Question

2 2 2

4

4 1 5cos

4

42128.1

Question A vector makes angles of and , respectively, with the - and - axes.

What is the angle between and the -ax

60 105 x

is?

y

z

u

u

2 2 2

2 2

cos cos cos 1

cos 1 cos cos

60 105

60 1

34 1

0

46

5

or

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)

n mOP OA OB

m n m n

������������������������������������������If AP:PB = m:n then

2 42, 1,4 3,1,7

4 2 4 21 2

2, 1,4 3,1,73 38 1 18

, ,3 3 3

OP

��������������

Question

UnderstandingConsider collinear points P, Q, and R. consider also a reference point O. Write OQ as a linear combination of OP and OR if:

a) Q divides PR (externally) in the ratio -3:8

Q P R

0

-3 5

3

5PQ PR

����������������������������

OQ OP PQ ������������������������������������������

3

5OQ OP PR ������������������������������������������

3

5OQ OP PO OR ��������������������������������������������������������

3

5OQ OP OP OR ��������������������������������������������������������

3 3

5 5OQ OP OP OR ��������������������������������������������������������

8 3

5 5OQ OP OR ������������������������������������������

Since:

Then:

8

Question

Find the projection of on 1,3,5 1,3, 2u v

2

2Proj

1,3,5 1,3, 21,3, 2

1,3, 2

21,3, 2

141 3 2

, ,7 7 7

u v vu vo

vn

�

2

coscos

1

cos

Pr v

u vu

v

u v

v

u v

v

u v v

v v

u v v

v

oj u

Multiply by unit vector in direction of v.

Question Find the point of intersection of: 1 3

3 1 2

13 1

2 1 3

x y z

x y z

Write in parametric form: 3

1

2 3

x s

y s

z s

2 13

1

3

x t

y t

z t

Therefore: 3 2 13

1 1

2 3 3

s t

s t

s t

9

7

s

t

27

8

21

x

y

z

Solve first two Check with third

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).

Question

Direction vector : 5,2d

Point : ,0 2, 3 3, 6

1 1

2 2

1,0

2

x t

t x

Vector Equation: 1, ,0 5,2

2x y t

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).

Question

Direction vector : 1 21,2,3 , 6, 1,2d d

Normal :

1 2

1, 2,3 6, 1, 2

7,16, 13

n d d

Scalar Equation:

7 16 13 0

7 2 16 3

7 16 1

13 4

3 0 0

0

1

0

1

x y z D

D

x

D

y z

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

2 2 2

Ax By Cz Dd

A B C

2 22

3 2 1 4

3 2 1

3 14

1

2

4

1 1

Question Determine the distance from the point P(2, -3, 2) to the line

0P A vd

v

��������������

1 1 1

3 2 4

x y z

First we need point on line: (1, -1, 1)

Now vector between two points:

1 2, 1 3 ,1 2 1,2, 1

2 2 2

1,2, 1 3,2,4

3 2 4

10,7,4

165

29

29

Question Determine the distance between two parallel planes x-y+z-10=0 and x-y+z+4=0

2 2 2

Ax By Cz Dd

A B C

2 22

1 1 1 10

1

4 0 0

1 1

14 3

3

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

Question Determine whether the following points are collinear: A(3,1,0), B(1,0,2), C(5, 2, -2)

(1,0, 2) (3,1,0)

( 2, 1,2)

AB OB OA

������������������������������������������

Find direction vector AB and AC and see if they are parallel:

(5, 2, 2) (3,1,0)

(2,1, 2)

AC OC OA

������������������������������������������

Therefore AB=-AC, thus AB and AC are parallel

Since they both share the common point A, then A, B, C are collinear

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)

(2,1,3) (1, 1,1) 4,1, 3

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): 4 32 1 1 0

6D

D

4 3 6 0x y z Therefore:

Question Given a line :

12

3 5

x z ky

And a plane: 2x-y-z+10=0

3,1,5 2, 1, 1 0

Determine the value k such that the line cuts the plane: a) at one single pointb) At many pointsc) 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

Determine the area of the parallelogram below, using the cross product.

6

10

400

2

sin

6 10 sin 40

38.6 [ ]

area

units

u v

u v

Question

2 3 4 1

1

2 2

x y z

x y z

x y z

Determine the point of intersection of the three planes:

Question

2 3 4 1

1 1 1 1

1 2 1 2

1 1 1 1

2 3 4 1

1 2 1 2

1 1 1 1

0 1 6 3

0 1 0 3

1 1 1 1

0 1 0 3

0 1 6 3

1 0 1 4

0 1 0 3

0 0 6 0

1 0 1 4

0 1 0 3

0 0 1 0

1 0 0

0 1 0

0 0 1

4

3

0

The point of intersection is: (-4,-3,0)