Vectors and Matrices

Question 1

How can vectors be written?

Answer 1

xi + yj or xi + yj + zk
(x) or (x)
(y)     (y)
         (z)
AB

Question 2

How to calculate the magnitude of a vector

Answer 2

sqrt(x^2 + y^2) or sqrt(x^2 + y^2 + z^2)

Question 3

How to work out a unit vector

Answer 3

(x/magnitude)
(y/magnitude)
(z/magnitude)

Question 4

Magnitude notation

Answer 4

|(x)| or |xi + yj + zk|
|(y)|
|(z)|
|AB|

Question 5

How can equation of lines be written?

Answer 5

Position Vector + constant(Direction Vector)

x) (3) (1
(y) = (2) + λ (3)
(z) (4) (4)

Question 6

How can you write the equation of a line in cartesian form given the vector form?

Answer 6

Form equations for each co-ordinate and re-arrange for λ or the constant. Then equate the equations.

Question 7

How to find the equation of a line given 2 points?

Answer 7

Points a and b.
Direction = b - a, position = a.
So a + λ(b - a).

Question 8

How to find if two lines are the same, intersect or parallel (2D)

Answer 8

If direction vectors are multiples of each other then parallel or same.
If they share a point (use a position vector) then same,
If intersect, equate the equations for x and y in both.
Solve simultaneous equations for λ and constant, plug values in for co-ordinates.

Question 9

How to find if two lines are the same, intersect, skew or parallel (3D)

Answer 9

Same as 2D for parallel or same.
Solve simultaneous equations for λ and other constant, then put into z equation.
If z equation works, then there’s an intersection, so plug values back in.
If it doesn’t work, lines are skew.

Question 10

Name the 2 dot product formulae for lines with direction vectors (a) (x)

                         (b) and (y)
                         (c) (z)

Answer 10

Calculating the dot product: ax + by + cz
cos(θ) = a . b/(|a||b|)
θ = angle between 2 vectors where both tips are touching, or both tails are touching of the vectors.

Question 11

What are 2 lines in relation to each other if their dot product is 0?

Answer 11

Perpendicular.

Question 12

How do you find the shortest distance between a point and a line?

Answer 12

Any point on line can be written using equation of line, so can write co-ordinates of point as (a + λd, b + λe, c + λf)
Find equation of line between that and point, take the direction vector.
Equation of line between that and position vector = direction vector.
Their dot product = 0, so solve for λ, then substitute to find point, then solve for magnitude.

Question 13

How to find the shortest distance between 2 points

Answer 13

Points (a, b, c) and (x, y, z)

sqrt((x - a)^2 + (y - b)^2 + (z - c)^2).

Question 14

What should you always use in a vector question?

Answer 14

Diagram drawn as 2D (even if 3D).

Question 15

How can you do dot product calculations on a calculator?

Answer 15

Optn
Mat/Vct F2
F6 x 2
DotP( F2
Exit x 2
Math (F4)
Mat/Vct F1
Put vector 1 in and comma in the middle, then vector 2

Question 16

Adding Matrices

Answer 16

Add corresponding elements from each matrix, matrices must have same dimensions.

Question 17

Multiplying

Answer 17

Must have same number of elements in each row in one as columns in the other.
Multiply the first term in the row of matrix 1 by the 1st element of the 1st column of matrix 2. Then do the 2nd element in the row and second element in the column, and add all the products. Then put this sum in the corresponding element in the result matrix, so if it is the top row and middle column, top middle element.

Question 18

Basic Matrix rules

Answer 18

AB != BA
A(BC) = (AB)C
A(B + C) = AB + AC
A0 = 0
AI = A = IA
A - A = 0

Question 19

What are the names of the 2 matrix rules:
A(BC) = (AB)C
A(B + C) = AB + AC

Answer 19

A(BC) = (AB)C - Associative

A(B + C) = AB + AC - Distributive

Question 20

How do you find the transformation matrix from the transformation?

Answer 20

Consider where (1, 0) and (0, 1) would go, and the co-ordinates are your left and right columns. Or, (1, 0, 0), (0, 1, 0) and (0, 0, 1).

Question 21

How do you find the transformation from the transformation matrix?

Answer 21

Check to see if the matrix is a rotation matrix or an enlargement (whether parallel to 1 axis, for all axes or on a plane), and then if neither, see if it’s a reflection.

Question 22

What’s the 2D rotation matrix?

Answer 22

(cosθ -sinθ)

sinθ cosθ

Question 23

What are the matrices for reflection in the planes x = 0, y = 0 and z = 0?

Answer 23

(x 0 0)
(0 y 0)
(0 0 z)
Where if plane x, x = -1, and the others are 1.
Where if plane y, y = -1, and the others are 1.
Where if plane z, z = -1, and the others are 1.

Question 24

What are the matrices for rotation in 3D around the axes x, y and z respectively?

Answer 24

(1 0 0 )
(0 cosθ -sinθ)
(0 sinθ cosθ)

(cosθ 0 sinθ)
(0 1 0 )
(-sinθ 0 cosθ)

(cosθ -sinθ 0)
(sinθ cosθ 0)
(0 0 1)

Question 25

How can you multiply or add matrices on a calculator?

Answer 25

Math (F4)
Mat/Vct (F2)
Select vector or choose vector with set dimensions, then put in operator (x, +, -, whatever) and then select other vector.

Question 26

If you know a combination transformation matrix is a combination of a rotation followed by an enlargement, how can you find the magnitude of the enlargement and θ, the angle of rotation?

Answer 26

k = length of line = square root of the sum of the squares of all the elements in the columns.
(a b) sqrt(a^2 + b^2)
(c d) sqrt (c^2 + d^2)
Divide matrix by k and solve for θ.