Vector equations of lines, skew lines and the angle between lines

Last time

Before we start: last time, in A definite integral that defines a constant, and solving it by iteration, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

A line in space

A line on a page needs a gradient and a point. A line in three dimensional space needs the same two things: a point to start from, and a direction to travel in. Once you can write that down, you can ask whether two lines meet, miss, or run side by side, and what angle they make.

The vector equation

The line through the point with position vector a, in the direction d, has vector equation r equals a plus lambda times d. The letter a is any known point on the line. The vector d gives the direction. The number lambda slides you along. Write r equals on the left. Cambridge withholds the mark if the r is missing.

Through two points

Take A with position vector 2 i plus 4 k, and B with position vector 5 i plus j plus 6 k. The direction is B minus A. Use A as the starting point. The line is r equals 2 i plus 4 k, plus lambda times, 3 i plus j plus 2 k. You may use B as the point instead. The equation looks different but is the same line.