(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.
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.
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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