Vectors

Question 1

3D vector equation

Answer 1

( a ) ( a1 )
r = ( b ) + λ ( b1 )
( c ) ( c1 )
r = position vector of a point of a line + λ(direction)

Question 2

Simplifying a 3D vector

Answer 2

( a + λa1 )
r = ( b + λb1 )
( c + λc1 )

Question 3

Vector equation of a line that passes through A and B

Answer 3

—->
AB = B - A
—->
r = A + λ(AB)

Question 4

Show a vector can be rewritten in a different way

Answer 4

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 5

Finding the points on a line a distance from a point

Answer 5

Simplify the vector in terms of a + λa1…
Use pythagoras on the x,y and z and square
Make a quadratic in terms of λ
Solve for λ and substitute into the original vector equation

Question 6

Vector in Cartesian form

Answer 6

Rearrange each In terms of λ
x - a1 y-a2. z-a3
——— = ——— = ———
b1. b2. c2

Question 7

Cartesian to vector

Answer 7

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

Question 8

Equation of a plane

Answer 8

r = a + λb + μc

Where A is a point on the plane and b is the direction vector from A to B and c the direction vector from A to C

Question 9

Cartesian equation of a plane

Answer 9

n1x + n2y + n3z = p

Question 10

Scalar/dot product

Answer 10

a.b

Multiply the x coefficients together, repeat with y and z then sum

Question 11

Angle between two vectors

Answer 11

a.b
cos θ = ———–
|a||b|
Take the mod of the whole thing if you want an acute angle

Question 12

How to see if two vectors are perpendicular

Answer 12

a.b = 0

Question 13

Finding a possible vector

Answer 13

Set z = 1, solve for x and y then scale

Question 14

Angle between two lines

Answer 14

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

Question 15

Equation of a plane

Answer 15

r.n = p
Where p = a.n
r is the position vector of a point on a plane and n is a normal to the plane
Write n as a vector and p as a scalar

Question 16

Cartesian equation of a plane

Answer 16

Set r = (x,y,z) and rearrange to equal 0

anx + bny + cnz - p = 0

Question 17

Angle between a line and a plane

Answer 17

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

Question 18

Angle between two planes

Answer 18

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

Question 19

Skew Lines

Answer 19

Two lines which do not meet

Question 20

Find the point where two lines intersect

Answer 20

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

Question 21

Intersection of a line and a plane

Answer 21

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 22

Shortest distance between two lines

Answer 22

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 23

Shortest distance between a point and a line

Answer 23

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 24

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

Answer 24

|n|

Question 25

Reflections of points in planes

Answer 25

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 26

Reflecting lines in planes

Answer 26

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 27

Finding a plane that contains a line and a point

Answer 27

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 28

Plane vector to Cartesian

Answer 28

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