Points, ratios and lines

The position vector of a point P(x, y, z) is the arrow from the origin O to P: OP = xî + yĵ + zk̂.

The vector from P to Q: 'end minus start'. PQ = OQ – OP. For P(1, 2, 3) and Q(4, 5, 6): PQ = 3î + 3ĵ + 3k̂.

Section formula: if R divides the line joining A (position vector a) and B (position vector b) in the ratio m : n,

• internally (R between A and B): r = (mb + na)/(m + n);
• externally (R outside the segment): r = (mb – na)/(m – n);
• midpoint (m = n = 1): (a + b)/2.

Notice the cross pairing: m goes with b and n goes with a.

Example: a = î + 2ĵ, b = 4î + 5ĵ. The point dividing AB internally in 2 : 1 is (2b + a)/3 = (9î + 12ĵ)/3 = 3î + 4ĵ.

Collinear points: A, B, C lie on one line if AB is a number times BC (they are parallel and share the point B). Equivalent test: |AB| + |BC| = |AC|.

Triangle law: AB + BC = AC. Going A to B then B to C is the same journey as going straight from A to C. This is how vectors are added, and why the lengths do not simply add: |a + b| is usually less than |a| + |b|.