Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — vector between two points
A is the point (1,2) and B is the point (4,6). Find AB and its magnitude.
- 1Write the position vectors: OA=(12) and OB=(46)A position vector runs from the origin to the point.
- 2AB=OB−OADestination minus start — the arrow runs to B.
- 3=(34)3 right and 4 up gets you from A to B.
- 4∣AB∣=32+42=25=5Pythagoras. This is also the distance between the two points.
Example 2 — multiplying two matrices
Given A=(2314) and B=(1052), find AB.
- 1Check the orders: 2×2 times 2×2 works and gives a 2×2.The inner numbers must match.
- 2Top-left: row 1 of A with column 1 of B: (2)(1)+(1)(0)=2Row across, column down.
- 3Top-right: row 1 with column 2: (2)(5)+(1)(2)=12
- 4Bottom-left: row 2 with column 1: (3)(1)+(4)(0)=3
- 5Bottom-right: row 2 with column 2: (3)(5)+(4)(2)=23
- 6AB=(231223)BA would give (176218) — order matters.
Example 3 — combining vectors
Given a=(23) and b=(5−1), find 2a−b.
- 12a=(46)A scalar multiplies both components.
- 2Now subtract componentwise.Tops with tops, bottoms with bottoms.
- 3Top: 4−5=−1
- 4Bottom: 6−(−1)=6+1=7Subtracting a negative adds — the usual place to slip.
- 52a−b=(−17)