9. Vectors

Question 1

What is r = a + λb?

Answer 1

Vector equation of straight line passing through point A with position vector a and parallel to vector b where λ is a scalar parameter

Question 2

What is r = c + λ(d - c)?

Answer 2

Vector equation of straight line passing through points C and D with position vectors c and d respectively where λ is a scalar parameter

Question 3

a = (a1 a2 a3)
b = (b1 b2 b3)
Equation of line with vector equation is r = a + λb

What is the Cartesian form of the equation of the line?

Answer 3

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

where each of these expressions is equal to λ

Question 4

What is r = a + λb + μc?

Answer 4

Vector equation of a plane

r is a position vector of a general point in the plane
a is the position vector of a point in the plane
b and c are non-parallel, non-zero vectors in the plane
λ and μ are scalars

Question 5

What is the Cartesian equation of a plane in 3 dimensions?

Answer 5

ax + by + cz = d

a, b, c and d are constants
(a b c) is normal vector to plane

Question 6

What is the scalar product of two vectors a and b?

Answer 6

a. b (“a dot b”)
a. b = |a| |b| cos θ is the definition of the scalar (dot) product

a and b must be facing away from a point
θ is the angle between a and b

Question 7

If a and b are position vectors of points A and B how do you work out the angle between them?

Answer 7

cos (AOB) = (a.b) / |a| |b|

Question 8

When are the non-zero vectors a and b perpendicular?

Answer 8

a.b = 0

Question 9

What is true about the dot product of a and b if they are parallel?

Answer 9

a.b = |a| |b|

so a.a = |a| ^ 2

Question 10

a = a1 i + a2 j + a3 k
b = b1 i + b2 j + b3 k

What is their scalar product?

Answer 10

a.b = (a1 a2 a3) . (b1 b2 b3)

= a1 b1 + a2 b2 + a3 b3

Question 11

What equation is the acute angle between 2 intersecting straight lines given by?

Answer 11

cos θ = |(a.b) / (|a| |b|)|

a and b are direction vectors of lines

Question 12

What is the scalar product form of the equation of a plane?

Answer 12

r.n = k

where k = a.n for any point in the plane with position vector a and n is the direction vector normal to the plane

Question 13

What formula gives the acute angle θ between the line r = a + λb and the plane with equation r.n = k?

Answer 13

sin θ = |(b.n) / (|b| |n|)|

Question 14

What formula gives the acute angle between the plane with equation r.n1 = k1 and the plane with equation r.n2 = k2?

Answer 14

cos θ = |(n1.n2) / (|n1| |n2|)|

Question 15

What are the two characteristics of skew lines?

Answer 15

1. They are not parallel

2. They do not intersect

Question 16

What is the unique line segment AB where A lies on l1 and B lies on l2 and l1 and l2 are non-intersecting

Answer 16

For any two non-intersecting lines l1 and l2 there is a unique line segment AB such that A lies on l1 and B lies on l2 and AB is perpendicular to both lines

Question 17

How is a perpendicular line drawn from point P to the line l?

Answer 17

A line drawn from P at right angles to the line l

Question 18

How is a perpendicular line drawn from point P to a plane Π?

Answer 18

A line is drawn from P parallel to the normal vector n

Question 19

What is shown in this equation r.n ̂ = k?

Answer 19

k is the length of the perpendicular from the origin to the plane Π where the equation of the plane is written in the form of r.n ̂ = k

n ̂ is a unit vector perpendicular to Π

Question 20

What is shown in the equation |aα+bβ+cγ-d|/√(a^2+b^2+c^2)?

Answer 20

The perpendicular distance from the point with co-ordinates (α, β, γ) and the plane with equation ax + by + cz = d

Question 21

What is meant by collinear?

Answer 21

If points all lie on the same straight line

Question 22

What is meant by coplanar?

Answer 22

If points all lie on the same plane

Question 23

What are the results for parallel unit vectors?

Answer 23

i. i = j.j = k.k = 1

i. j = i.k = j.i = j.k = k.i = k.j = 0

Question 24

What do modulus signs ensure?

Answer 24

The angles are acute