Further vectors
The scalar product and angles
Multiply two vectors and get a number that measures how much they point the same way. Use it to find any angle in three dimensions, and to test for right angles in a single line of working.
≈ 35 minThe vector equation of a line
One point and one direction pin down a line. Write it as , slide along it by changing , test whether points lie on it, and find the angle between two lines from their directions alone.
≈ 35 minIntersecting, parallel and skew lines
In a plane, two lines either cross or are parallel. In three dimensions they can also miss each other completely. Learn the three-step test that decides, and how to find the crossing point when there is one.
≈ 40 minFoot of the perpendicular and shortest distances
The closest point on a line to a given point is where a perpendicular meets it. Find it with one scalar product, then get distances, reflections and areas almost for free.
≈ 40 min