(9) Can we multiply 3 vectors together?
The following answers to the 4 questions at the end of the previous section
show the various different possibilities of what can happen.
First question
(2i . 3i) . 4j = ?
This one is impossible to find because working out the
dot product of (2i . 3i)
gives the number 6.
We now can't find 6 . 4j because the dot product is defined
to be a way of multiplying 2 vectors.
RULE
Suppose a, b and c are 3 vectors.
It is always impossible to find (a . b) . c because
a . b
gives a number. The dot product of this number with the vector c
does not exist.
Second question
(2j . 4j) x 3k = ?
This one is also impossible because working out 2j . 4j gives
the number 8.
We now can't find 8 x 3k because the
cross product is defined to be a way of multiplying two
vectors.
RULE Suppose a, b and
c are 3 vectors.
It is always impossible to find (a . b) x c.
Third question
(2i x 3j) x 4i = ?
Working out 2i x 3j gives
the vector answer 6k.
Now, working out 6k x 4i gives the vector 24j, so we
have
(2i x 3j) x 4i = 24j.
We have used i x j = k
and k x i = j. These
results come in the special cases
section near the end of the
vector product section.
RULE Suppose that a, b
and c are 3 vectors.
We can always work out the vector answer
to (a x b) x c.
This is called the
vector triple product.
Fourth question
(2i x 3j) . 4k = ?
This is an example of what's called a scalar
triple product.
Because it has a neat geometrical interpretation it gets a whole section
to itself.
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