Vectors (checked)

Question 1

Show a vector can be rewritten in a different way

Answer 1

Show one of the directions is another multiplied by a scalar (parallel)
Let λ = 0 and find the μ value that gives the same point to find a common point

Question 2

Cartesian to vector

Answer 2

Set each part of the Cartesian form equal to λ and rearranged for x,y, z
Write back into Cartesian form

Question 3

How to see if two vectors are perpendicular

Answer 3

a.b = 0

Question 4

Angle between two lines

Answer 4

a.b
cos θ = ———–
|a||b|
Where a and b are the direction vectors of two lines

Question 5

If the scalar product of an angle is obtuse how does that affect the angle

Answer 5

The angle between the vectors will be obtuse.

Question 6

Angle between a line and a plane

Answer 6

Find the angle between the line and the normal and do 90 - θ

Question 7

Angle between two planes

Answer 7

Find the angle between the two normals and subtract from 180, take whichever acute answer you get

Question 8

Skew Lines

Answer 8

Two lines which do not meet

Question 9

Find the point where two lines intersect

Answer 9

Write as column vectors, solve the first two equations simultaneously for μ and λ then substitute into the third to check

Question 10

Intersection of a line and a plane

Answer 10

Write the line as a column vector and substitute as r in the plane, solve for λ using the dot product then substitute back in

Question 11

Shortest distance between two lines

Answer 11

1) Group together for A and B
2) Carry out B-A for AB
3) Take the dot product with each direction and set to 0
4) Solve simultaneously for μ and λ
5) Substitute into AB and take the modulus

Question 12

Shortest distance between a point and a line

Answer 12

1) Group together each component of the line’s vector
2) Subtract the point from the line
3) Set the dot product of that and the line’s direction to 0
4) Use that to find λ
5) Substitute and take the modulus

Question 13

Shortest distance between point x and plane r.n = d

Answer 13

|n|

Question 14

Reflections of points in planes

Answer 14

1) Write a line as r = point + λ(direction of normal)
2) Substitute that as r in the plane equation
3) Solve for λ
4) Substitute 2λ in place of λ in r = point + λ(direction of normal)

Question 15

Reflecting lines in planes

Answer 15

1) Write the vector equation of the line
2) Substitute as r in the plane equation and find the point of intersection
3) Reflect the point where λ=0 in the plane
4) Find the line which passes through those two points - don’t use λ

Question 16

Finding a plane that contains a line and a point

Answer 16

Let λ=0 and then -1 to find another 2 points
Write as simultaneous equations equalling 1 and solve
Scale so x,y,z are integers
Write in the correct form

Question 17

Plane vector to Cartesian

Answer 17

Use x,y,z = a + bλ + cμ
Use the y,z equations to find μ and λ in terms of y and z
Substitute into the x equation