Vector Product of two vectors

To describe how a plane is tilted (rather than just a line, as slope does), two vectors lying in the plane

are combined to produce a third vector perpendicular to that plane -- this is the vector (cross) product.

Definition. For non-collinear, non-zero vectors $\bar a,\bar b$, choose the unit vector $\hat n$

perpendicular to the plane of $\bar a,\bar b$ by the right-hand rule: curl the right-hand fingers from

$\bar a$ towards $\bar b$ through the angle $\theta$ between them, and the thumb then points along $\hat n$.

Then

Remarks.

1. $|\bar a\times\bar b|=|\bar a||\bar b|\sin\theta$ (since $|\hat n|=1$).
2. $\bar a\times\bar b$ is perpendicular to the plane of $\bar a$ and $\bar b$.
3. The unit vector along $\bar a\times\bar b$ is $\hat n=(\bar a\times\bar b)/|\bar a\times\bar b|$.
4. If $\bar a,\bar b$ are coplanar but non-collinear, any vector $\bar c$ in space can be written $\bar c=x\bar a+y\bar b+z(\bar a\times\bar b)$, since $\bar a,\bar b,\bar a\times\bar b$ together span the whole of space.
5. $\bar b\times\bar a=-(\bar a\times\bar b)$ (anti-commutative): reversing the order reverses the sense of the right-hand curl, flipping $\hat n$ to $-\hat n$.
6. $\bar a\times\bar b=\bar 0$ exactly when $\bar a,\bar b$ are collinear (either is zero, or $\sin\theta=0$); in particular $\bar a\times\bar a=\bar 0$ always, and if $\bar a=k\bar b$ then $\bar a\times\bar b=\bar 0$.
7. Distributive laws: $\bar a\times(\bar b+\bar c)=\bar a\times\bar b+\bar a\times\bar c$ and $(\bar b+\bar c)\times\bar a=\bar b\times\bar a+\bar c\times\bar a$.
8. For scalars $m,n$: $(m\bar a)\times\bar b=m(\bar a\times\bar b)=\bar a\times(m\bar b)$ and $(m\bar a)\times(n\bar b)=mn(\bar a\times\bar b)$.
9. $\hat\imath\times\hat\imath=\hat\jmath\times\hat\jmath=\hat k\times\hat k=\bar 0$, and, since $\hat\imath,\hat\jmath,\hat k$ form a right-handed triplet, $\hat\imath\times\hat\jmath=\hat k,\ \hat\jmath\times\hat k=\hat\imath,\ \hat k\times\hat\imath=\hat\jmath$. Component formula. For $\bar a=a_1\hat\imath+a_2\hat\jmath+a_3\hat k,\ \bar b=b_1\hat\imath +b_2\hat\jmath+b_3\hat k$,

   $\bar a\times\bar b=\begin{vmatrix}\hat\imath&\hat\jmath&\hat k\\ a_1&a_2&a_3\\ b_1&b_2&b_3\end{vmatrix}.$

   Angle from the cross product. Since $|\bar a\times\bar b|=|\bar a||\bar b|\sin\theta$, $\sin\theta=|\bar a\times\bar b|/(|\bar a||\bar b|)$, giving a second route (besides the dot product) to the angle between two vectors -- useful particularly when a right angle or the vectors' sine is more natural than their cosine. Geometric meaning (area). If $\bar a,\bar b$ share an initial point, they determine a parallelogram of base $|\bar a|$ and height $|\bar b|\sin\theta$, so its area is $|\bar a|\cdot|\bar b|\sin\theta=|\bar a\times\bar b|$. Worked examples.

• For $\bar a=\hat\imath+\hat\jmath-\hat k,\ \bar b=2\hat\imath+4\hat\jmath+6\hat k$: expanding the determinant gives $\bar a\times\bar b=10\hat\imath-8\hat\jmath+2\hat k$, and dotting this with $\bar a$ and with $\bar b$ separately both give $0$, confirming it is orthogonal to both.
• To find all vectors of a given magnitude perpendicular to the plane of two given vectors: compute $\bar a\times\bar b$, find its magnitude, form the unit vector $\hat n=(\bar a\times\bar b)/|\bar a\times\bar b|$, and scale $\pm\hat n$ by the required magnitude.
• If $\bar u+\bar v+\bar w=\bar 0$, crossing this equation with $\bar v$ (and separately noting $\bar v\times\bar v=\bar 0$) gives $\bar u\times\bar v=-\bar w\times\bar v=\bar v\times\bar w$; a similar manipulation shows $\bar v\times\bar w=\bar w\times\bar u$ too, so all three pairwise cross products are equal.
• Showing $\bar a\times(\bar b\times\bar c)\ne(\bar a\times\bar b)\times\bar c$ in general: compute both sides numerically for specific vectors and observe they differ, confirming the cross product is not associative.
• Area of a triangle with vertices $A,B,C$: form $\overrightarrow{AB},\overrightarrow{AC}$, take $\left|\overrightarrow{AB}\times\overrightarrow{AC}\right|$ and halve it.
• Area of a parallelogram $KLMN$: form two adjacent side vectors from one vertex and take the magnitude of their cross product directly (no halving).
• $|\bar u\times\bar v|=|\bar u||\bar v|\sin\theta$ is evaluated directly once $|\bar u|,|\bar v|,\theta$ are all known or read off a sketch.
• Distributive law verification: for specific $\bar a,\bar b,\bar c$, compute $\bar a\times(\bar b+\bar c)$ and $\bar a\times\bar b+\bar a\times\bar c$ separately and check they agree, term by term.
• Showing three points with given position vectors are collinear: compute $\overrightarrow{AB}$ and $\overrightarrow{AC}$ and show $\overrightarrow{AB}\times\overrightarrow{AC}=\bar 0$ (rather than checking one is a scalar multiple of the other component-by-component); since $\bar 0$ cross product means the two vectors are collinear, and $A$ is shared, $A,B,C$ are collinear.
• Finding a unit vector perpendicular to two given vectors $\overrightarrow{PQ},\overrightarrow{PR}$, and the sine of the angle between them: form the cross product, divide by its magnitude for the unit vector, and use $\sin\theta=|\overrightarrow{PQ}\times\overrightarrow{PR}|/(|\overrightarrow{PQ}||\overrightarrow{PR}|)$ for the sine.
• Given $|\bar a|,|\bar b|,|\bar a\times\bar b|$, first solve $|\bar a\times\bar b|=|\bar a||\bar b|\sin\theta$ …