How Do You Find the Vector Product of Two Vectors?

The vector product (also known as the cross product) of two vectors results in a new vector that is perpendicular to both original vectors. Here's how to find it:

Understanding the Vector Product

The vector product, denoted as a × b, produces a vector c with the following properties:

• Magnitude: The magnitude of c is given by |c| = |a| |b| sin(θ), where |a| and |b| are the magnitudes of vectors a and b, respectively, and θ is the angle between them.
• Direction: The direction of c is perpendicular to both a and b. The right-hand rule determines the specific direction: point the fingers of your right hand in the direction of a, curl them towards b, and your thumb will point in the direction of c.

Methods for Calculating the Vector Product

There are two primary methods for calculating the vector product:

1. Using Components (Determinant Method)

This method is most suitable when you know the components of the vectors. If a = (a_x, a_y, a_z) and b = (b_x, b_y, b_z), then a × b is calculated as follows:

a × b = ( (a_yb_z - a_zb_y), (a_zb_x - a_xb_z), (a_xb_y - a_yb_x) )

This can be more easily remembered using a determinant:

a × b = | i j k |
| a_x a_y a_z |
| b_x b_y b_z |

Where i, j, and k are the unit vectors along the x, y, and z axes, respectively. Expanding the determinant gives the same component-wise formula.

Example:

Let a = (1, 2, 3) and b = (4, 5, 6).

a × b = ( (2*6 - 3*5), (3*4 - 1*6), (1*5 - 2*4) ) = ( (12 - 15), (12 - 6), (5 - 8) ) = (-3, 6, -3)

2. Using Magnitude and Angle

If you know the magnitudes of the vectors (|a| and |b|) and the angle (θ) between them, you can find the magnitude of the vector product using:

|a × b| = |a| |b| sin(θ)

You still need to determine the direction of the resulting vector c using the right-hand rule. This method is less computationally intensive than the component method when you already know the magnitudes and angle.

Example:

Let |a| = 5, |b| = 3, and θ = 30 degrees.

|a × b| = 5 3 sin(30°) = 15 * (1/2) = 7.5

You would then need to use the right-hand rule to determine the direction of the resulting vector, which will be perpendicular to both a and b.

Key Properties of the Vector Product

• Anti-commutative: a × b = - (b × a)
• Distributive: a × (b + c) = a × b + a × c
• Not Associative: a × (b × c) ≠ (a × b) × c
• If a and b are parallel (θ = 0° or 180°), then a × b = 0 (the zero vector).

Applications

The vector product has numerous applications in physics and engineering, including:

• Calculating torque
• Determining the magnetic force on a moving charge
• Finding the area of a parallelogram defined by two vectors