(10) The scalar triple product

and what it means geometrically

We start by looking at the numerical example of

(2i x 3j) . 4k = ?

This particular scalar triple product gives the volume of a rectangular box. Here's how the working out goes.
First, we have the cross product of 2i x 3j = 6k.
The picture below shows you that this 6k gives the size of the area of the yellow base of the left-hand box, but as a vector in the k direction.
finding the volume of these boxes

Now, finding the dot product, we have 6k . 4k = 24, so giving the actual volume of the left-hand box.
Similarly, (3j x 4k) . 2i gives the volume of the right-hand box. This is the same size but we are starting with the yellow left-hand face to find its volume. We have
(3j x 4k) . 2i = 12i . 2i = 24.

RULE Suppose we have three vectors a, b and c.
We can always find the scalar triple product of (a x b) . c and the answer will be a number, not a vector.

This answer will give the volume of the slant-sided box made by sliding the 3 vectors a, b and c together. (This box is called a parallelepiped.)
I show how this works in the drawing below.

how the scalar triple product gives a volume

We have a x b = |a||b| sin R u = Au
where u is a unit vector perpendicular to the yellow parallelogram made by a and b and A is its area.
So (a x b) . c = Au . c = A|u||c| cos S
but |u| = 1 and |c| cos S = h which is the perpendicular height of the box.
Therefore (a x b) . c = the volume of the box whose edges are the vectors a, b and c.

Equally, (b x c) . a and (c x a) . b give the volume of the same box, so we have

(a x b) . c = (b x c) . a = (c x a) . b

Being able to work out dot and cross products makes it possible to fully explore how we can write down vector equations for lines and planes. The next section deals with how we can find the vector equation of a line.


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