Cross Product Formula
The Formula
When to use: Place two arrows flat on a table. The cross product points straight up from the table, perpendicular to both. Its length tells you how much area the two arrows span—like the area of a parallelogram with the arrows as sides. If the arrows are parallel, they span no area, so the cross product is the zero vector.
Quick Example
\mathbf{a} \times \mathbf{b} = \langle 0, 0, 1 \rangle
The result points along the z-axis, perpendicular to both \mathbf{a} and \mathbf{b}.
Notation
What This Formula Means
The cross product of two 3D vectors \mathbf{a} = \langle a_1, a_2, a_3 \rangle and \mathbf{b} = \langle b_1, b_2, b_3 \rangle is a new vector \mathbf{a} \times \mathbf{b} that is perpendicular to both \mathbf{a} and \mathbf{b}. Its magnitude equals the area of the parallelogram formed by \mathbf{a} and \mathbf{b}.
Place two arrows flat on a table. The cross product points straight up from the table, perpendicular to both. Its length tells you how much area the two arrows span—like the area of a parallelogram with the arrows as sides. If the arrows are parallel, they span no area, so the cross product is the zero vector.
Formal View
Worked Examples
Example 1
mediumSolution
- 1 Step 1: Use the formula: \mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle.
- 2 Step 2: = \langle 0 \cdot 0 - 0 \cdot 1, 0 \cdot 0 - 1 \cdot 0, 1 \cdot 1 - 0 \cdot 0 \rangle.
- 3 Step 3: = \langle 0, 0, 1 \rangle.
- 4 Check: \hat{\mathbf{i}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}} by the right-hand rule ✓
Answer
Example 2
hardCommon Mistakes
- Swapping the order and forgetting the sign change: \mathbf{a} \times \mathbf{b} \neq \mathbf{b} \times \mathbf{a}
- Trying to compute a cross product of 2D vectors without extending them to 3D (append 0 as the third component)
- Sign errors in the component formula—the middle component has a subtracted order (a_3 b_1 - a_1 b_3, not a_1 b_3 - a_3 b_1)
Why This Formula Matters
Cross products find normal vectors to surfaces, compute torque and angular momentum in physics, determine the orientation of three points (left or right turn), and calculate areas in 3D geometry.
Frequently Asked Questions
What is the Cross Product formula?
The cross product of two 3D vectors \mathbf{a} = \langle a_1, a_2, a_3 \rangle and \mathbf{b} = \langle b_1, b_2, b_3 \rangle is a new vector \mathbf{a} \times \mathbf{b} that is perpendicular to both \mathbf{a} and \mathbf{b}. Its magnitude equals the area of the parallelogram formed by \mathbf{a} and \mathbf{b}.
How do you use the Cross Product formula?
Place two arrows flat on a table. The cross product points straight up from the table, perpendicular to both. Its length tells you how much area the two arrows span—like the area of a parallelogram with the arrows as sides. If the arrows are parallel, they span no area, so the cross product is the zero vector.
What do the symbols mean in the Cross Product formula?
\mathbf{a} \times \mathbf{b} uses the multiplication sign. Can also be computed as a 3 \times 3 determinant: \mathbf{a} \times \mathbf{b} = \det\begin{bmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{bmatrix}.
Why is the Cross Product formula important in Math?
Cross products find normal vectors to surfaces, compute torque and angular momentum in physics, determine the orientation of three points (left or right turn), and calculate areas in 3D geometry.
What do students get wrong about Cross Product?
The cross product is only defined for 3D vectors (and 7D, but that is rarely encountered). Also, order matters: \mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a}).
What should I learn before the Cross Product formula?
Before studying the Cross Product formula, you should understand: dot product, vector operations, determinant.