FM - Core Pure 1 - 9) Vectors

Question 1

Vector equation of a straight line

Answer 1

r = a + λb

Question 2

Cartesian form of a vector equation of a line

Answer 2

(x-a₁)/b₁ = (y-a₂)/b₂ = (z-a₃)/b₃ = λ

Question 3

Vector equation of a plane

Answer 3

r = a + λb + µc

Question 4

Cartesian form of a vector equation of a plane

Answer 4

ax + by + cz = d → (a b c) is the normal vector to the plane, and d is a·n

Question 5

Formula for the scalar (dot) product and information that shows when two vectors are perpendicular and parallel

Answer 5

• a·b = |a||b|cos(θ)
• Perpendicular → a·b = 0
• Parallel → a·b = |a||b| → a·a = |a|²

Question 6

Scalar product from of a vector equation of a plane

Answer 6

r.n = k or r.n = a.n

Question 7

Formula for the acute angle between two lines, a line and a plane, and two planes

Answer 7

• Two lines: cos(θ) = |(a·b /|a||b|)|
• A line and a plane: sin(θ) = |(b·n /|b||n|)|
• Two planes: cos(θ) = |(n₁·n₂ /|n₁||n₂|)|
  → when both vectors are pointing in the same direction - if they point in opposite directions, do 180° - angle

Question 8

Method to find the point of intersection of two lines in vectors

Answer 8

1. Write each equation in a single column and set them equal to each other
2. Write out the three linear equations
3. Try to solve first two simultaneously (if no solutions, they don’t intersect)
4. Test solutions on third equation (if it doesn’t satisfy, they don’t intersect)
5. Substitute values back in to either equation of line to find point of intersection

Question 9

Method to find the point of intersection of a line and a plane in vectors

Answer 9

1. Write (a+λb) · (n) = k
2. Solve for λ
3. Substitute that value back into λ to find point of intersection

Question 10

Meaning of two lines being skew

Answer 10

• They don’t intersect and aren’t parallel
• Direction vectors aren’t multiples of each other

Question 11

Where is the shortest distance between vectors

Answer 11

The line that is perpendicular to the vector (dot product = 0)