VECTORS

Question 1

Define scalar product

Answer 1

a.b = IaIIbI cos ø

Question 2

Define vector product

Answer 2

a x b = IaIIbI sinø ñ

Question 3

What does i x i = ?

Answer 3

j x j = k x k = 0

Question 4

What does

a) i x j = ?
b) k x j = ?

Answer 4

a) k
b) -i

Question 5

Relationship between vector product and parallel vectors

Answer 5

If a x b = 0, the two vectors are parallel

Question 6

Vector product, area of triangle

Answer 6

1/2 I a x b I

Question 7

Vector product area of parallelogram

Answer 7

I a x b I

Question 8

What does a. b x c mean?

Answer 8

Scalar triple product,

a. (b x c)

Question 9

Importance of order in scalar triple product

Answer 9

Same cyclic order as vector product

a. b x c = b. c x a

Question 10

What does a x (b + c) mean? (Distributive law)

Answer 10

a x b + a x c

Question 11

Volume of a pyramid

Answer 11

1/3 I a. b x c I

Question 12

Volume of a tetrahedron

Answer 12

1/6 I a. b x c I

Question 13

Volume of a triangular prism

Answer 13

1/2 I a. b x c I

Question 14

What does coplanarity mean?

How can it be proven?

Answer 14

If vectors are co-planar, they lie on the same plane

I a. b x c I = 0

Question 15

Parametric form of equation of a line

Answer 15

r = a + µb

Question 16

Cartesian/Direction ratio form of equation of a line

Answer 16

x - a1/b1 = y - a2/b2 = z - a3/b3 (=µ)

a from parametric equation = specific point

b = direction

Question 17

Change to cartesian form of equation of line when direction vector contains a 0

Answer 17

No denomiator, comma instead of = sign

Question 18

Direction cosines and the angle a parallel line makes with x, y, z axes

using line x - a1/b1 = y - a2/b2 = z - a3/b3

Answer 18

vector b is parallel

angles it makes with the three axes : cos ø =

b_i/IbI = direction cosines, denoted as l, m and n

Question 19

Sum of square of direction vectors = ?

Answer 19

l² + m² + n² = 1

Question 20

Vector equation form of equation of line

Answer 20

( r - a ) x b = 0

(Because they are parallel)

Question 21

Equation of plane using two non-parallel vectors on plane

Answer 21

r = a + ßb + µc

b and c are the two non-parallel vectors, a is a specific point on the plane.

Question 22

Scalar product equation of a plane

Answer 22

r . n = d

d = a . n

a is a point on the plane

n is a vector perpendicular to plane, found by x-ing two direction vectors

Question 23

Cartesian equation of plane

Answer 23

n₁x + n₂y + n₃z = d

Question 24

Angle between a line and a plane

Answer 24

90^. - the acute angle (180 - angle if not acute) between the line and the normal to the plane calculated using:

cosø = I b .n / IbI InI I

b is direction vector of line

Question 25

Angle between two planes

Answer 25

Scalar product using the two normals: I **n₁ . n₂ / In₁I In₂I** I

Question 26

Shortest distance from point p to line AB?

Answer 26

The perpendicular, using vector product: I AP x AB I/ IABI A and B are point on the line

Question 27

Shortest distance between two lines

Answer 27

* Find any vector **AB** between two points, one on each line * x them to find perpendicular vector * calculate I **AB**. **n** I / I **n** I

Question 28

Shortest distance point to plane

Answer 28

* Find equation of line that goes through point A perpendicular to the plane * Find the point of intersection between point on the line and the plane * Find the distance AP

Question 29

Intersection of planes x + 2y - z = 2 and 3x -y + 2z = 1 With description

Answer 29

Line of intersection is perpendicular to the normals of both planes the two normals here are 1 3 2 -1 -1 2 x these to get which is the direction vector 3 - 5 - 7 Substitute values for x, y and z into the the equations of the planes to get a specific point

Question 30

Intersection between line **r** = ( 2, -3, 4) + µ(-1, 2, -3) and plane 2x - 3y + z = 6 With description

Answer 30

**r** = (x, y, z) x = 2- µ y = -3 + 2µ z = 4 -3µ put these in equation of plane, to get value for µ and then put this into the equation of the plane