Q1.7 Multiplying vectors: the cross (or vector) product (H)

Write down the definition of the cross product of two vectors.

The three vectors $\vec{A}$, $\vec{B}$ and $\vec{C}$ form a right-angled triangle in which B=2 and C=1 (in arbitrary units).

  1. Determine the values of

    $\vec{A} \times \vec{B}$ and $\vec{B} \times \vec{C}$.

  2. Compare $\vec{A} \times \vec{B}$ with $\vec{B} \times \vec{A}$ and comment.
  3. Interpret your results geometrically.
Graphic - No title - ABCtriangle
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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

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Solution

From Key Point 0.4 the cross product of two vectors ($\vec{D}$ and $\vec{E}$ 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.

  1. It follows that in this case

    \[ \mid\vec{A}\times\vec{B}\mid = AB \sin{\theta_{AB}} =AB \frac{C}{A}= BC =2 \]

    The direction (get your corkscrew out and see which way it goes when you turn $\vec{A}$ into $\vec{B}$) is into the plane.

    Similarly

    \[ \mid\vec{B}\times \vec{C}\mid = BC \sin{\theta_{BC}} =2 \]

    Again the direction is into the plane.

  2. While $\vec{A}\times\vec{B}$ is into the plane $\vec{B}\times\vec{A}$ is out of the plane: you have to turn the corkscrew anticlockwise to take $\vec{B}$ into $\vec{A}$; 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.
  3. 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 $\vec{A}$ and $\vec{B}$. You can read off its area (base times altitude) as BC=2, in accord with the above result.
    Graphic - No title - parallelogram

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.