Q1.7 Multiplying vectors: the cross (or vector) product (H)
Write down the definition of the cross product of two vectors.
The three vectors ,
and
form a right-angled triangle
in which B=2 and C=1 (in arbitrary units).
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Hint
Reveal

Hint
You’ll need to remind yourself of Key Point 0.4...and how to use a corkscrew.
Does the order of the vectors matter in the cross product?
Solution
Reveal

Solution
From Key Point 0.4
the cross product of two vectors (
and
say) is a vector of magnitude
DEsinθ
where θ is the smaller angle between the two vectors.
The direction is given by the right handed corkscrew rule.
It follows that in this case
The direction (get your corkscrew out and see which way it goes when you turn
into
) is into the plane.
Similarly
Again the direction is into the plane.
- While
is into the plane
is out of the plane: you have to turn the corkscrew anticlockwise to take
into
; and when you do so it comes out of the plane. Thus, in contrast to the scalar product, the order of the two vectors you are multiplying does matter in the case of the vector product: switch the order and you switch the sign.
- Geometrically the magnitude of the cross product is
the area of the parallelogram you can construct from its sides.
The dashed line shows the parallelogram built from vectors and
. You can read off its area (base times altitude) as BC=2, in accord with the above result.
If you haven’t met cross products before, you’ll probably find this a bit daunting. Don’t worry. We won’t make much use of cross products in this course. But if you take a physics course next year you’ll bump into them much more often.