The Cross Product

Today’s opening problem was physical: a wrench on a stubborn bolt. Where do you push, and which way, to turn it hardest? Push along the handle and nothing turns at all; push perpendicular to it, out at the end, and the bolt gives. The turning effect — torque — depends on two vectors, and it is one: it points along the bolt’s axis, perpendicular to both the handle and your push. What multiplication takes two vectors and produces a third, perpendicular to both? That is the cross product — the product that points.

$\vec{a}\times \vec{b}=(a_2{b}_3-a_3{b}_2,a_3{b}_1-a_1{b}_3,a_1{b}_2-a_2{b}_1)$

The formula looks like a tongue-twister; the pattern is a cycle ($1\to 2\to 3\to 1$), and it exists only in three-space. For $\vec{a}=(2,0,0)$ and $\vec{b}=(0,3,0)$ — one vector along each of the first two axes — the product is $(0,0,6)$, straight up the third. Two directions in, the missing direction out.

Self-check: is $(0,0,6)$ really perpendicular to

both? Test with the tool from The Dot Product: $(2,0,0)\cdot (0,0,6)=0$ and $(0,3,0)\cdot (0,0,6)=0$. Both zero, both perpendicular. This check works on every cross product you will ever compute — dot your answer with each original vector, and anything nonzero means an arithmetic slip. A self-marking computation is a gift; take it every time.

The magnitude is an area

The direction is perpendicular; the size has its own meaning:

$|\vec{a}\times \vec{b}|=|\vec{a}||\vec{b}|\sin \theta$

That $\sin \theta$ is precisely the height factor of the parallelogram with sides $\vec{a}$ and $\vec{b}$ — so the magnitude of the cross product is that parallelogram’s area. Where the dot product’s $\cos \theta$ rewarded agreement, $\sin \theta$ rewards perpendicularity: the cross product of parallel vectors is the zero vector (no parallelogram at all), and the wrench turns hardest when your push meets the handle at $90°$.

The properties earned an Always, Sometimes, Never round: $\vec{a}\times \vec{b}=\vec{b}\times \vec{a}$ — never, unless the result is zero; reversing the order flips the answer’s direction. Not commutative — the first product you have ever met that cares about order, which is exactly why it can encode orientation. Dot and Cross Product Practice has areas, torque, and normals; Equations of Planes is where “a vector perpendicular to two others” stops being a curiosity and becomes the whole method.

Curriculum connection

C2.6

perform the operation of cross product on two vectors represented in Cartesian form in three-space [i.e., using $\vec{a}\times \vec{b}=(a_2{b}_3-a_3{b}_2,a_3{b}_1-a_1{b}_3,a_1{b}_2-a_2{b}_1)$], determine the magnitude of the cross product (i.e., using $|\vec{a}\times \vec{b}|=|\vec{a}||\vec{b}|\sin \theta$), and describe applications of the cross product (e.g., determining a vector orthogonal to two given vectors; determining the turning effect [or torque] when a force is applied to a wrench at different angles) Sample problem: Explain how you maximize the torque when you use a wrench and how the inclusion of a ratchet in the design of a wrench helps you to maximize the torque.

Link to original

C2.7

determine, through investigation, properties of the cross product (e.g., investigate whether it is commutative, distributive, or associative; investigate the cross product of collinear vectors) Sample problem: Investigate algebraically the relationship between the cross product of the vectors $\vec{a}=(1,0,1)$ and $\vec{b}=(0,1,-1)$ and the cross product of scalar multiples of $\vec{a}$ and $\vec{b}$. Does this relationship apply to any two vectors?

Link to original

C2.8

solve problems involving dot product and cross product (e.g., determining projections, the area of a parallelogram, the volume of a parallelepiped), including problems arising from real-world applications (e.g., determining work, torque, ground speed, velocity, force) Sample problem: Investigate the dot products $\vec{a}\cdot (\vec{a}\times \vec{b})$ and $\vec{b}\cdot (\vec{a}\times \vec{b})$ for any two vectors $\vec{a}$ and $\vec{b}$ in three-space. What property of the cross product $\vec{a}\times \vec{b}$ does this verify?

Link to original