Vectors and Matrices 1

Section 1Scalars and vectors

A scalar has size only, such as mass, time, or temperature. A vector has size and direction, such as displacement, velocity, or force. Easy to mix up - speed is a scalar, but velocity is a vector because it includes direction.

Section 2Vector notation

A vector can be written as a column, such as (3, 4) in column form, meaning 3 units across and 4 units up. The top number usually represents horizontal movement, and the bottom number represents vertical movement. Negative components show movement left or down.

Section 3Position vectors

A position vector gives the location of a point from the origin. If point A is (5, -2), its position vector is (5, -2) in column form. In plain words, it tells you how to travel from (0, 0) to the point.

Section 4Vector operations

To add vectors, add matching components. To subtract vectors, subtract matching components. To multiply a vector by a scalar, multiply every component by that scalar.

Section 5Magnitude

The magnitude of a vector is its length. For vector (a, b), magnitude = sqrt(a^2 + b^2). This is just Pythagoras working on the horizontal and vertical components.

Section 6Matrices

A matrix is a rectangular arrangement of numbers in rows and columns. The order of a matrix is rows by columns. A 2 by 3 matrix has 2 rows and 3 columns.

Section 7Matrix operations

Matrices can be added or subtracted only when they have the same order. Add or subtract matching entries. For scalar multiplication, multiply every entry in the matrix by the scalar.