Cross Product

Cross Product

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Prerequisites

Determinant
Dot Product

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What is Cross Product

1. What is Cross Product?

For two vectors in $\mathbb{R}^3$:

\[v = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix},\qquad w = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix}\]

The cross product is:

\[v \times w =\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{vmatrix}\]

Expanding:

\[\hat{i}(v_2 w_3 - v_3 w_2) - \hat{j}(v_1 w_3 - v_3 w_1) + \hat{k}(v_1 w_2 - v_2 w_1)\]

Each component is a $2\times2$ determinant.

2. Magnitude and Orientation

Magnitude

The magnitude of the cross product is:

\[\|v \times w\| = \|v\| \|w\| \sin\theta\]

This equals the area of the parallelogram formed by $v$ and $w$.

So:

• More perpendicular → $\sin\theta$ large → area large
• Similar direction → $\sin\theta$ small → area small
• Same direction → area = 0

$ \text{Area of parallelogram} = \text{base} \times \text{height} = |\vec{v}|(|\vec{w}|\sin\theta) ,\quad |\vec{v} \times \vec{w}| = \text{Area} $

Orientation

In 2D, we can think of a signed area:

\[\det \begin{bmatrix} v_1 & w_1 \\ v_2 & w_2 \end{bmatrix} = v_1 w_2 - v_2 w_1\]

This equals the z-component of $v \times w$.

Interpretation:

• If $v$ is counterclockwise (left) of $w$ → positive
• If $v$ is clockwise (right) of $w$ → negative

So orientation determines sign.

3. Basis vector Example

\[\hat{i} \times \hat{j} = \hat{k}\]

Right-hand rule:

• Index fingers along $\hat{i}$
• Middle toward $\hat{j}$
• Thumb points in $+\hat{k}$ direction

So:

\[\hat{i} \times \hat{j} = +\hat{k}\]

But:

\[\hat{j} \times \hat{i} = -\hat{k}\]

Cross product is not commutative.

4. Required Properties of a Cross Product

A binary vector operation satisfying all these properties exists naturally only in 3-dimensional space (and a special case in 7 dimensions).

A valid cross product must satisfy the following properties:

Property	Meaning
Bilinearity	Linear in both vector inputs
Anticommutativity	Reversing order flips the sign
Orthogonality	Result is perpendicular to both vectors
Magnitude Rule	Magnitude equals parallelogram area
Orientation	Direction determined by right-hand rule

4.1 Bilinearity

The operation must be linear in each argument.

\[(a+b) \times c = a \times c + b \times c\] \[a \times (b+c) = a \times b + a \times c\] \[(\lambda a) \times b = \lambda (a \times b)\] \[a \times (\lambda b) = \lambda (a \times b)\]

4.2 Anticommutativity

\[a \times b = -(b \times a)\]

4.3 Orthogonality

The result must be perpendicular to both input vectors.

\[(a \times b) \cdot a = 0\] \[(a \times b) \cdot b = 0\]

4.4 Magnitude Condition

The magnitude of the cross product must equal the area of the parallelogram formed by the vectors.

\[|a \times b| = |a||b|\sin\theta\]

4.5 Orientation (Right-Hand Rule)

The direction of the cross product follows the right-hand rule.

If the fingers of the right hand rotate from (a) toward (b), the thumb points in the direction of

\[a \times b\]

This defines the orientation of the resulting vector.

5. Determinant

\[\text{Area scaling by matrix } A = |\det(A)|\]

Original area × determinant = new area.

If a matrix transforms two vectors:

\[A v \times A w\]

Its magnitude equals:

\[|\det(A)| \cdot \|v \times w\|\]

6. Scalar Triple Product

For three vectors (a, b, c \in \mathbb{R}^3), the scalar triple product is defined as

\[(a \times b) \cdot c = \det \begin{pmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{pmatrix} = signed\; volume(V)\]

The computation occurs in two steps:

1. Compute the cross product $a \times b$
2. Take the dot product of the result with $c$

The result is a scalar value.

The scalar triple product connects three important geometric operations:

Operation	Meaning
Dot Product	Projection
Cross Product	Area
Scalar Triple Product	Volume

7. What if the Dimension is over 3?

NOT Cross product -> Wedge product