Chapter 9: Vectors

Question 1

What is the vector equation of a straight line?

Answer 1

r = a + λb, r being a general point on the line, a being a position vector, λ being a scalar and b being a direction vector.

Question 2

What is a scalar?

Answer 2

A multiplier for the direction vector. It can be given as any Greek letter, most common are lambda (λ) and mu(μ).

Question 3

What is the vector equation of a straight line going through points C and D?

Answer 3

r = c + λ(d-c)

Question 4

What is the Cartesian form of the equation of a straight line if a = |a₁| and b = |b₁| |a₂|
| b₂| |a₃|
| b₃|?

Answer 4

(x-a₁)/b₁=(y-a₂)/b₂=(z-a₃)/b₃

Question 5

What is the vector equation of a plane?

Answer 5

r = a + λb + μc, r being a general point on the line, a being a position vector, λ being a scalar and b and c being non-parallel direction vectors.

Question 6

What is the Cartesian equation of a plane, with |a|
|b|
|c|?

Answer 6

ax + by + cz = d

Question 7

What is the scalar product?

Answer 7

The scalar product, or the dot product, is used to determine the angle between 2 vectors.

Question 8

What is the formula for the scalar product?

Answer 8

a.b = |a||b|cosθ

Question 9

What is the dot product of a=|a₁| and b=|b₁|
|a₂| |b₂|
|a₃| |b₃|?

Answer 9

a.b = a₁b₁ + a₂b₂ + a₃b₃

Question 10

What is the formula for the acute angle between 2 intersecting lines when a and b are direction vectors?

Answer 10

cosθ = |(a.b/|a||b|)|