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Week 3 Discussion Notes

Table of Contents

Cross Product

While it might look like a random definition, the cross product is the "best" way to find a vector perpendicular to two vectors. Geometrically, the cross product looks like the following:

The direction of v⃗×w⃗\vec{v} \times \vec{w} is determined by the right-hand rule, like the axes. If you look at the picture, you'll see that order matters! v⃗×w⃗\vec{v} \times \vec{w} and w⃗×v⃗\vec{w} \times \vec{v} point in opposite directions, and you can check that the directions are consistent with the right-hand rule.

Algebraically, you can calculate the cross product with the following determinant formula:

v⃗×w⃗=∣i^j^k^v1v2v3w1w2w3∣=∣v2v3w2w3∣i^−∣v1v3w1w3∣j^+∣v1v2w1w2∣k^=(v2w3−v3w2)i^−(v1w3−v3w1)j^+(v1w2−v2w1)k^.\begin{aligned} \vec{v} \times \vec{w} &= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{vmatrix} \\ &= \begin{vmatrix} v_2 & v_3 \\ w_2 & w_3 \end{vmatrix} \hat{i} - \begin{vmatrix} v_1 & v_3 \\ w_1 & w_3 \end{vmatrix} \hat{j} + \begin{vmatrix} v_1 & v_2 \\ w_1 & w_2 \end{vmatrix} \hat{k} \\ &= \p{v_2w_3 - v_3w_2} \hat{i} - \p{v_1w_3 - v_3w_1} \hat{j} + \p{v_1w_2 - v_2w_1} \hat{k}. \end{aligned}

Notice that there is an additional minus sign when calculating the j^\hat{j}-coordinate. Lastly, on #6 of this week's worksheet, you will show that the length of v⃗×w⃗\vec{v} \times \vec{w} is the area of parallelogram spanned by v⃗\vec{v} and w⃗\vec{w}:

The area of the parallelogram is the height (∥w⃗∥sin⁡θ\norm{\vec{w}} \sin\theta) times the base length (∥v⃗∥\norm{\vec{v}}), which is what #6 tells you:

∥v⃗×w⃗∥=∥v⃗∥∥w⃗∥sin⁡θ,\norm{\vec{v} \times \vec{w}} = \norm{\vec{v}} \norm{\vec{w}} \sin\theta,

where θ\theta is the angle between v⃗\vec{v} and w⃗\vec{w} measured in the interval [0,π]\br{0, \pi} (or [0∘,180∘]\br{0^\circ, 180^\circ} in degrees).

Triple Product

I made a GeoGebra graph to help show why u⃗⋅(v⃗×w⃗)=0\vec{u} \cdot \p{\vec{v} \times \vec{w}} = 0 tells you that u⃗,v⃗,w⃗\vec{u}, \vec{v}, \vec{w} are coplanar.