(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?
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
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
I've now added an extra page on equations of planes.
equations of planes in the form Ax + By + Cz = D
or back to the previous section
or back to the vectors home page.