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Vector Algebra and Analytic Geometry of Space

The Vector Product of Vectors (the Cross Product)

The Definition of the Vector Product

Vector product

Let a⃗=(a1,a2,a3)\vec{a} = (a_1, a_2, a_3) and b⃗=(b1,b2,b3)\vec{b} = (b_1, b_2, b_3). The vector product is defined as:

a⃗×b⃗=(a2b3−a3b2,a3b1−a1b3,a1b2−a2b1)\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)

The result of the vector product is a vector, and it is also called the cross product.

How to Compute It

The vector product can be computed with a determinant:

The determinant form of the vector product
a⃗×b⃗=∣i⃗j⃗k⃗a1a2a3b1b2b3∣\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

The Geometric Meaning

The geometric meaning of the vector product

The resulting vector is perpendicular to the plane spanned by a⃗\vec{a} and b⃗\vec{b}, and its direction follows the right-hand rule:

∣a⃗×b⃗∣=∣a⃗∣∣b⃗∣sin⁡θ|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin\theta

where θ\theta is the angle between the two vectors.

The Right-Hand Rule

With the four fingers turning from a⃗\vec{a} towards b⃗\vec{b}, the direction of the thumb is the direction of a⃗×b⃗\vec{a} \times \vec{b}.

Properties

The basic properties of the vector product

Anti-commutativity: a⃗×b⃗=−b⃗×a⃗\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}

Distributivity: a⃗×(b⃗+c⃗)=a⃗×b⃗+a⃗×c⃗\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}

Associativity with scalar multiplication: (ka⃗)×b⃗=k(a⃗×b⃗)(k\vec{a}) \times \vec{b} = k(\vec{a} \times \vec{b})

The cross product of a vector with itself: a⃗×a⃗=0⃗\vec{a} \times \vec{a} = \vec{0}

Applications

Testing for Parallelism

The condition for two vectors to be parallel
a⃗∥b⃗⇔a⃗×b⃗=0⃗\vec{a} \parallel \vec{b} \Leftrightarrow \vec{a} \times \vec{b} = \vec{0}

Computing an Area

The area of a parallelogram

The area of the parallelogram with a⃗\vec{a} and b⃗\vec{b} as adjacent sides is:

S=∣a⃗×b⃗∣S = |\vec{a} \times \vec{b}|

Finding a Normal Vector

The normal vector of a plane

The cross product of two vectors a⃗\vec{a} and b⃗\vec{b} gives the normal vector of the plane:

n⃗=a⃗×b⃗\vec{n} = \vec{a} \times \vec{b}

Computing a Moment of Force

The moment-of-force formula
M⃗=r⃗×F⃗\vec{M} = \vec{r} \times \vec{F}

where r⃗\vec{r} is the vector from the pivot to the point of application, and F⃗\vec{F} is the force vector.

Exercises

Exercise 1

Find the vector product of a⃗=(1,2,3)\vec{a} = (1, 2, 3) and b⃗=(4,0,−1)\vec{b} = (4, 0, -1).

the vector productcomputing a cross productthe determinant

Approach:

Use the coordinate formula for the vector product.

Steps:

  1. a⃗×b⃗=(2×(−1)−3×0,3×4−1×(−1),1×0−2×4)\vec{a} \times \vec{b} = (2 \times (-1) - 3 \times 0, 3 \times 4 - 1 \times (-1), 1 \times 0 - 2 \times 4)
  2. =(−2−0,12+1,0−8)=(−2,13,−8)= (-2 - 0, 12 + 1, 0 - 8) = (-2, 13, -8)

Answer: a⃗×b⃗=(−2,13,−8)\vec{a} \times \vec{b} = (-2, 13, -8)

Exercise 2

Decide whether the vectors a⃗=(1,2,3)\vec{a} = (1, 2, 3) and b⃗=(2,4,6)\vec{b} = (2, 4, 6) are parallel.

parallel vectorsthe vector productthe zero vector

Approach:

Compute the vector product; if the result is the zero vector, the two vectors are parallel.

Steps:

  1. a⃗×b⃗=(2×6−3×4,3×2−1×6,1×4−2×2)\vec{a} \times \vec{b} = (2 \times 6 - 3 \times 4, 3 \times 2 - 1 \times 6, 1 \times 4 - 2 \times 2)
  2. =(12−12,6−6,4−4)=(0,0,0)= (12 - 12, 6 - 6, 4 - 4) = (0, 0, 0)

Answer: the vector product is the zero vector, so the two vectors are parallel.

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
a⃗×b⃗\vec{a} \times \vec{b}Vector productcross productThe vector product of the vectors a\mathbf{a} and b\mathbf{b}
i⃗,j⃗,k⃗\vec{i}, \vec{j}, \vec{k}Basis vectorsbasis vectorsThe standard basis vectors
sin⁡θ\sin\thetaSinesine thetaThe sine of the angle
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