Product of two vectors
Product of Two Vectors
In vector mathematics, there are multiple ways to multiply vectors together, each with its own geometric interpretation and applications. The two most common products are the dot product (also known as the scalar product) and the cross product.
Dot Product (Scalar Product)
The dot product of two vectors results in a scalar (hence the name scalar product) and is a measure of the vectors' mutual alignment. It is defined for two vectors A and B as follows:
Formula
$$ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos(\theta) $$
where:
- $|\mathbf{A}|$ is the magnitude of vector A
- $|\mathbf{B}|$ is the magnitude of vector B
- $\theta$ is the angle between the vectors
If the vectors are given in Cartesian coordinates, the dot product can be calculated as:
$$ \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z $$
for 3-dimensional vectors A = $(A_x, A_y, A_z)$ and B = $(B_x, B_y, B_z)$.
Properties
- Commutative: $\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$
- Distributive: $\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}$
- Scalar multiplication: $(c\mathbf{A}) \cdot \mathbf{B} = c(\mathbf{A} \cdot \mathbf{B})$ for any scalar c
Example
Given two vectors A = (1, 2, 3) and B = (4, -5, 6), their dot product is:
$$ \mathbf{A} \cdot \mathbf{B} = (1)(4) + (2)(-5) + (3)(6) = 4 - 10 + 18 = 12 $$
Cross Product (Vector Product)
The cross product of two vectors results in a vector that is perpendicular to the plane containing the original vectors. It is defined for vectors A and B in three-dimensional space as follows:
Formula
$$ \mathbf{A} \times \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \sin(\theta) \mathbf{n} $$
where:
- $|\mathbf{A}|$ and $|\mathbf{B}|$ are the magnitudes of vectors A and B
- $\theta$ is the angle between the vectors
- $\mathbf{n}$ is a unit vector perpendicular to the plane containing A and B, following the right-hand rule
In Cartesian coordinates, the cross product can be calculated using the determinant of a matrix:
$$ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \ \end{vmatrix} $$
Properties
- Anticommutative: $\mathbf{A} \times \mathbf{B} = -(\mathbf{B} \times \mathbf{A})$
- Distributive: $\mathbf{A} \times (\mathbf{B} + \mathbf{C}) = \mathbf{A} \times \mathbf{B} + \mathbf{A} \times \mathbf{C}$
- Scalar multiplication: $(c\mathbf{A}) \times \mathbf{B} = c(\mathbf{A} \times \mathbf{B})$ for any scalar c
Example
Given two vectors A = (1, 2, 3) and B = (4, -5, 6), their cross product is:
$$ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 2 & 3 \ 4 & -5 & 6 \ \end{vmatrix} = (2 \cdot 6 - 3 \cdot -5)\mathbf{i} - (1 \cdot 6 - 3 \cdot 4)\mathbf{j} + (1 \cdot -5 - 2 \cdot 4)\mathbf{k} = (12 + 15)\mathbf{i} - (6 - 12)\mathbf{j} - (5 + 8)\mathbf{k} = (27, 6, -13) $$
Comparison Table
| Property | Dot Product | Cross Product |
|---|---|---|
| Result | Scalar | Vector |
| Commutative | Yes | No (Anticommutative) |
| Distributive | Yes | Yes |
| Defined in | Any dimension | 3 dimensions only |
| Geometric Meaning | Measures alignment | Gives perpendicular vector |
| Formula | $\mathbf{A} \cdot \mathbf{B} = | \mathbf{A} |
| Coordinates Formula | $\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z$ | $\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \ \end{vmatrix}$ |
Understanding the product of two vectors is crucial for many applications in physics, engineering, and computer graphics. The dot product is often used to determine the angle between vectors or to project one vector onto another, while the cross product is used to find a vector perpendicular to a plane or to calculate torque and angular momentum.