The Vector Cross Product

When the two input vectors are parallel, their cross product is the zero vector. This is because the angle $\theta$ between them is 0 or $\pi$, making $\sin \theta = 0$, and thus the magnitude $|\mathbf{a}||\mathbf{b}|\sin \theta$ is zero.

Conditions: The two vectors are parallel (or anti-parallel).; You are computing their cross product.

Unlike the scalar dot product, the cross product of two three-dimensional vectors is a vector. To compute it, you can use the determinant mnemonic, expanding its component formula with alternating signs.

Conditions: The vectors are three-dimensional.; You are using the determinant mnemonic to find the components.

The magnitude of the cross product of two vectors $\mathbf{a}$ and $\mathbf{b}$ with angle $\theta$ between them is $|\mathbf{a}||\mathbf{b}|\sin \theta$. Geometrically, this value represents the area of the parallelogram formed by the two vectors.

Conditions: $\mathbf{a}$ and $\mathbf{b}$ are three-dimensional vectors.; $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.

Reversing the operands reverses the sign of the resulting vector. Specifically, $\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}$.

Conditions: You are computing the cross product of two vectors.; You swap the order of the vectors.

A nonzero cross product result is perpendicular to both input vectors. Its specific direction is set by the right-hand rule.

Conditions: The cross product is nonzero (input vectors are not parallel).; You are determining the orientation of the resulting vector.

No, the cross product is not generally associative. This means that $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$ is not necessarily equal to $\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$.

Conditions: You are evaluating expressions involving multiple cross products.; You are checking if grouping affects the result.

Yes, the cross product distributes over addition. This means $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}$ and $(\mathbf{a} + \mathbf{b}) \times \mathbf{c} = \mathbf{a} \times \mathbf{c} + \mathbf{b} \times \mathbf{c}$.

Conditions: You are simplifying expressions involving cross products and sums.; You are applying algebraic properties to vector operations.