The vector product, also known as the cross product, is a fundamental operation in vector algebra that takes two vectors and produces a third vector.
What is the Vector Product?
Based on the reference, the vector product is defined as:
- A binary operation on two vectors in three-dimensional space.
- It results in a vector that is perpendicular to both vectors.
The Vector product of two vectors, a and b, is denoted by a × b. This operation produces a vector, unlike the scalar product (dot product) which produces a scalar. The resulting vector's direction and magnitude are specific and crucial to understanding its applications.
Think of it as creating a vector that 'points away' from the plane formed by the two input vectors, and whose length depends on how 'perpendicular' the input vectors are to each other.
Characteristics of the Vector Product
The vector product (a × b) possesses several key characteristics:
1. Resultant is a Vector
Unlike the dot product, the cross product of two vectors produces a vector.
2. Direction of the Resultant Vector
As stated in the reference, its resultant vector is perpendicular to a and b. This perpendicular vector is normal to the plane containing both vectors a and b.
- Right-Hand Rule: The specific direction of the resulting vector a × b is determined by the right-hand rule. If you curl the fingers of your right hand from vector a to vector b (through the smaller angle), your thumb points in the direction of a × b.
3. Magnitude of the Resultant Vector
The magnitude of the vector a × b is given by the formula:
|a × b| = |a| |b| sin(θ)
Where:
- |a| is the magnitude of vector a.
- |b| is the magnitude of vector b.
- θ is the angle between vectors a and b (0° ≤ θ ≤ 180°).
This means:
- If a and b are parallel (θ = 0° or 180°), sin(θ) = 0, so |a × b| = 0. The cross product of parallel vectors is the zero vector.
- If a and b are perpendicular (θ = 90°), sin(θ) = 1, so |a × b| = |a| |b|. The magnitude is maximized.
4. Properties
Here are some important properties of the vector product:
- Anti-commutative: The order of the vectors matters. Swapping the order reverses the direction of the resultant vector:
b × a = - (a × b) - Distributive over vector addition: The cross product distributes over addition:
a × (b + c) = (a × b) + (a × c) - Not Associative: In general, (a × b) × c ≠ a × (b × c).
- Scalar Multiplication: A scalar multiple can be factored out:
k(a × b) = (ka) × b = a × (kb) - Cross Product with Itself: The cross product of any vector with itself is the zero vector:
a × a = 0
5. Geometric Interpretation
The magnitude of the vector product |a × b| is equal to the area of the parallelogram spanned by vectors a and b.
Example
Let's consider the standard basis vectors in 3D space: i, j, and k.
- i is a unit vector along the x-axis.
- j is a unit vector along the y-axis.
- k is a unit vector along the z-axis.
Using the definition and right-hand rule:
- i × j = k (Resultant is perpendicular to x and y, magnitude is 11sin(90°)=1, direction is +z by RHR)
- j × i = -k (Anti-commutative property)
- j × k = i
- k × j = -i
- k × i = j
- i × k = -j
- i × i = 0 (Cross product with itself is zero)
For general vectors a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, the cross product can be calculated using a determinant:
a × b = (a₂b₃ - a₃b₂) i - (a₁b₃ - a₃b₁) j + (a₁b₂ - a₂b₁) k
Applications
The vector product has numerous applications in physics and engineering, such as:
- Calculating torque: Torque (τ) is given by τ = r × F, where r is the position vector and F is the force.
- Calculating angular momentum: Angular momentum (L) is given by L = r × p, where r is the position vector and p is the linear momentum.
- Finding the normal vector to a plane.
- Calculating the area of a triangle or parallelogram in 3D space.
In summary, the vector product is a crucial operation that yields a perpendicular vector and has significant geometric and physical interpretations.
