Vectors and Matrices 2

Section 1Vector geometry

Vector geometry uses vectors to prove facts about points and lines. If A and B are points, vector AB is found by subtracting the position vector of A from the position vector of B. Watch out: AB and BA point in opposite directions.

Section 2Position vectors and paths

A position vector tells where a point is from the origin. If OA = a and AB = b, then OB = a + b because you travel from O to A, then from A to B. This is useful when a diagram gives a path instead of coordinates.

Section 3Parallel and collinear vectors

Vectors are parallel when one is a scalar multiple of the other. Points are collinear when they lie on the same straight line. Here's the trick: if AB = kBC for some scalar k, then A, B, and C are on one line.

Section 4Midpoints and ratios

The midpoint of A(x1, y1) and B(x2, y2) is found by averaging the coordinates. In vector form, the midpoint position vector is 1/2(a + b). For other ratios, split the journey in the given parts instead of guessing from the picture.

Section 5Matrix multiplication

To multiply matrices, each entry comes from a row times a column. The number of columns in the first matrix must match the number of rows in the second matrix. Easy to mix up - matrix multiplication is not done by simply multiplying matching positions.

Section 6Transformation matrices

A 2 by 2 matrix can transform a point or vector by multiplying the matrix by its column vector. The result is the image point. For example, [[2, 0], [0, 2]] doubles both coordinates, so it is an enlargement scale factor 2 about the origin.

Section 7Common matrix transformations

Some matrices have special meanings. [[1, 0], [0, 1]] is the identity and leaves points unchanged. [[-1, 0], [0, 1]] reflects in the y-axis, [[1, 0], [0, -1]] reflects in the x-axis, and [[0, 1], [1, 0]] reflects in the line y = x.