Vectors and Matrices

Question 1

Scalar

Answer 1

Only has magnitude

Question 2

Vector

Answer 2

Has direction and magnitude

Question 3

Multiplying vectors

Answer 3

Multiply by a scalar constant

Question 4

Adding/subtracting vectors

Answer 4

Add/subtract x1 and x2, then y1 and y2

Question 5

Scalar product

Answer 5

Multiply corresponding coordinates and add up the results

Question 6

v ∙ v = ?

Answer 6

|v|^2

Question 7

u ∙ v = ?

Answer 7

|u||v| cos⁡ φ

Question 8

Commutativity

Answer 8

u ∙ v = v ∙ u

Question 9

Distributivity

Answer 9

u ∙ (v + w) = u ∙ v + u ∙ w

Question 10

(au) ∙ (bv) = ?

Answer 10

(ab)(u ∙ v)

Question 11

Linear combination

Answer 11

Sum of vectors by multiplying original vectors

Question 12

Matrix

Answer 12

A table consisting of rows and columns of numbers

Question 13

Square matrix

Answer 13

A matrix where the number of rows = number of columns

Question 14

Adding/subtracting matrices

Answer 14

Performed element by element

Remember that you can only add/subtract matrices with the same dimensions!

Question 15

Multiplication by a constant

Answer 15

All elements are multiplied by the constant

Question 16

Multiplication by a vector

Answer 16

Multiply row by row using scalar product

Remember that the vector must have the same number of elements as there are columns in the matrix!

Question 17

Matrix-by-matrix multiplication

Answer 17

Multiply element by element using scalar product (row in A by column in B)

Question 18

Position vector

Answer 18

The vector from the origin to a point

Question 19

Rotation matrix

Answer 19

To rotate p by an angle φ counter-clockwise about the origin, calculate p’ = Rp, where R is:
(cosφ -sinφ)
(sinφ cosφ)

Question 20

Identity matrix

Answer 20

Iv = v

Question 21

Inverse matrix

Answer 21

Given a square matrix A, if there is a matrix B such that BA = I, then B is the inverse of A

Question 22

How can inverse matrices be used to solve simultaneous equations?

Answer 22

o If A-1 is known, multiply both sides by it
o Matrix multiplication is associative, so can shift brackets
o A-1A = I by definition of inverse matrix
o Iv = v by definition of identity matrix