Geometrical Interpretation of Cross Product

The cross product, also known as the vector product, is an operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both of the original vectors. The cross product has significant applications in physics, engineering, and mathematics, particularly in the study of vectors, rotations, and three-dimensional geometry.

Definition

Given two vectors $\vec{a}$ and $\vec{b}$ in three-dimensional space, their cross product $\vec{a} \times \vec{b}$ is defined as:

$$ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_x & a_y & a_z \ b_x & b_y & b_z \ \end{vmatrix} $$

where $\hat{i}$, $\hat{j}$, and $\hat{k}$ are the unit vectors along the x, y, and z axes, respectively, and $a_x$, $a_y$, $a_z$, $b_x$, $b_y$, and $b_z$ are the components of vectors $\vec{a}$ and $\vec{b}$.

Geometrical Interpretation

The geometrical interpretation of the cross product involves both direction and magnitude:

• Direction: The resulting vector is perpendicular to the plane formed by the two original vectors, following the right-hand rule. If you point your index finger in the direction of $\vec{a}$ and your middle finger in the direction of $\vec{b}$, your thumb will point in the direction of $\vec{a} \times \vec{b}$.

• Magnitude: The magnitude of the cross product is equal to the area of the parallelogram with sides $\vec{a}$ and $\vec{b}$. Mathematically, it is given by:

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

where $\theta$ is the angle between vectors $\vec{a}$ and $\vec{b}$.

Important Points and Differences

Aspect	Description
Resultant Type	The cross product of two vectors is always a vector.
Direction	Perpendicular to the plane containing the original vectors, determined by the right-hand rule.
Magnitude	Equal to the area of the parallelogram formed by the two vectors.
Commutativity	The cross product is not commutative: $\vec{a} \times \vec{b} = -(\vec{b} \times \vec{a})$.
Zero Vector	If the cross product is a zero vector, the original vectors are parallel or one of them is a zero vector.

Examples

Example 1: Computing the Cross Product

Let's compute the cross product of vectors $\vec{a} = \langle 2, 3, 4 \rangle$ and $\vec{b} = \langle 5, 6, 7 \rangle$.

$$ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 2 & 3 & 4 \ 5 & 6 & 7 \ \end{vmatrix} = (3 \cdot 7 - 4 \cdot 6) \hat{i} - (2 \cdot 7 - 4 \cdot 5) \hat{j} + (2 \cdot 6 - 3 \cdot 5) \hat{k} = -3 \hat{i} + 6 \hat{j} - 3 \hat{k} $$

So, $\vec{a} \times \vec{b} = \langle -3, 6, -3 \rangle$.

Example 2: Geometrical Interpretation

Consider vectors $\vec{a} = \langle 1, 0, 0 \rangle$ and $\vec{b} = \langle 0, 1, 0 \rangle$. The cross product is:

$$ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & 0 & 0 \ 0 & 1 & 0 \ \end{vmatrix} = (0 \cdot 0 - 0 \cdot 1) \hat{i} - (1 \cdot 0 - 0 \cdot 0) \hat{j} + (1 \cdot 1 - 0 \cdot 0) \hat{k} = \hat{k} $$

The resulting vector $\hat{k}$ is perpendicular to both $\vec{a}$ and $\vec{b}$, and its magnitude is 1, which is the area of the parallelogram (in this case, a square with side length 1) formed by $\vec{a}$ and $\vec{b}$.

Example 3: Parallel Vectors

If $\vec{a} = \langle 1, 2, 3 \rangle$ and $\vec{b} = \langle 2, 4, 6 \rangle$, then $\vec{b}$ is a scalar multiple of $\vec{a}$, indicating that they are parallel. The cross product is:

$$ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & 2 & 3 \ 2 & 4 & 6 \ \end{vmatrix} = (2 \cdot 6 - 3 \cdot 4) \hat{i} - (1 \cdot 6 - 3 \cdot 2) \hat{j} + (1 \cdot 4 - 2 \cdot 2) \hat{k} = 0 \hat{i} - 0 \hat{j} + 0 \hat{k} = \vec{0} $$

The cross product is the zero vector, which confirms that the vectors are parallel.

Understanding the cross product's geometrical interpretation is crucial for visualizing vector operations and solving problems involving torque, angular momentum, and forces in three-dimensional space.