## (15) Angles between lines and planes

In this section we'll look at how to find
• the angle between 2 lines
• the angle between 2 planes
• the angle between a line and a plane.

Each of these will involve calculating the size of an angle between 2 vectors. How to do this is explained near the end of the dot product section.
Now all we need to do is to decide which 2 vectors we need to work with in each case.

### The angle between 2 lines

We define the angle between 2 lines to be the angle between their direction vectors placed tail to tail. Notice that this definition works equally well if the lines don't actually cut each other since we then just slide the 2 direction vectors together until their tails meet.
I show an example of this in the drawing below.

The two lines have the equations r = a + tb and r = c + sd.
The angle between the lines is found by working out the dot product of b and d.
We have b.d = |b||d| cos A.
From this, knowing b and d, we can find the angle A.
Taking the example I gave near the end of the dot product section,
suppose that b = 2i - 3j + k and d = 4i + j - 3k.
Then the working out there shows that the angle A is 84 degrees.

### The angle between 2 planes

It is important to choose the correct angle here. It is defined as the angle between 2 lines, one in each plane, so that they are at right angles to the line of intersection of the 2 planes (like the angle between the tops of the pages of an open book).
The picture below shows part of 2 planes and the angle between them.

To find this angle, will we first have to find the equation of the line of intersection of the 2 planes, and then find 2 vectors which are in the planes and perpendicular to this?
Fortunately no! We just need to know a normal vector to each of the planes. Then we can find the angle we want very neatly as I show in the drawing below.

The angle between the planes is the same as the angle between their 2 normal vectors (sliding their tails together if necessary).
Now we just use n.m = |n||m| cos A
and find the angle in the same way as we did for the 2 lines.

### The angle between a line and a plane

Again, the neatest method is to use a normal vector to the plane. I show how this works in the drawing below.

We slide the normal vector n until its tail is at the point of intersection with the line L with the plane P. Then n and L together define a plane which is perpendicular to plane P. The angle which line L makes with plane P is defined to be the red angle A in this plane.
Since A and B together make a right angle, we can find A by using the dot product of n and the direction vector b of line L to first find cos B.
Or we can find A even more directly by using the trig identity
cos B = cos (90 - A) = sin A
so n.b = |n||b| sin A so giving A.
(If your trig is a bit shaky, you will find that Chapter 5 in my book will help you. This link will tell you what topics it includes.)
I've now added an extra page on equations of planes.

equations of planes in the form Ax + By + Cz = D