Product of Vectors — Dot and Cross Product Guide

TL;DR

The product of vectors comes in two distinct forms: the dot product (\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta), which returns a number, and the cross product (\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta, \hat{n}), which returns a vector. This article maps both, shows when each one is the right tool, and extends the idea to the scalar triple product.

What Is the Product of Vectors?

The product of vectors is the operation of multiplying two vectors together, and it takes one of two forms depending on what you need to know. The dot product (also called the scalar product) multiplies two vectors and returns a single number. The cross product (also called the vector product) multiplies two vectors and returns a new vector perpendicular to both.

There is no third "ordinary" multiplication of vectors and no division of vectors at all. The reason is that a vector carries both size and direction, so "multiply" has to decide what to do with the directions. The dot product collapses the two directions into one number through the cosine of the angle between them; the cross product builds a brand-new direction out of them through the sine. Everything else in this article is detail hanging off that one fork.

What Are the Two Types of Product of Vectors?

The two types are the dot product and the cross product. They differ in three ways at once — the operator symbol, the trig function inside, and the type of answer that comes out.

Notice the last two rows mirror each other. The dot product vanishes exactly when the cross product is largest (perpendicular vectors), and the cross product vanishes exactly when the dot product is largest (parallel vectors). They are reading two complementary halves of the same geometric relationship.

The dot product, in brief

The dot product takes (\vec{a}) and (\vec{b}) and returns (|\vec{a}||\vec{b}|\cos\theta) — a number measuring how much one vector points along the other. In component form, for (\vec{a}=(a_1,a_2,a_3)) and (\vec{b}=(b_1,b_2,b_3)):

[\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3.]

Because (\cos 90° = 0), the dot product is zero precisely when the two vectors are perpendicular — which is why it's the standard test for a right angle.

The cross product, in brief

The cross product takes (\vec{a}) and (\vec{b}) and returns a vector of magnitude (|\vec{a}||\vec{b}|\sin\theta) pointing perpendicular to the plane the two vectors sit in, with direction set by the right-hand rule. In component form:

[\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1).]

When Do You Use the Dot Product vs the Cross Product?

Reach for the dot product when the question is about alignment — how much of one vector lies along another — and reach for the cross product when the question is about turning or area — something that needs a new direction.

• Use the dot product to find an angle.
• Use the dot product to test perpendicularity.
• Use the dot product for work and projection.
• Use the cross product for torque and rotation.
• Use the cross product for area.
• Use the cross product to find a perpendicular direction.

A quick way to keep the split straight: cosine measures togetherness, sine measures apartness. The dot product runs on cosine, so it peaks when the vectors agree; the cross product runs on sine, so it peaks when they're at right angles.

Examples of Product of Vectors

Example 1

Find the dot product of (\vec{a}=(2,3,1)) and (\vec{b}=(4,−1,5)).

[\vec{a} \cdot \vec{b} = (2)(4) + (3)(−1) + (1)(5) = 8 - 3 + 5 = 10.]

Example 2

A common slip — find the cross product of (\vec{a}=(1,2,0)) and (\vec{b}=(3,0,0)).

Wrong attempt. A student multiplies matching components and calls that the cross product. Check it against what a cross product must do: the answer is supposed to be perpendicular to both inputs, yet (3,0,0) is parallel to (\vec{b}), not perpendicular to it. The correct way uses the difference pattern:

[\vec{a} \times \vec{b} = ((2)(0)−(0)(0), (0)(3)−(1)(0), (1)(0)−(2)(3))=(0,0,−6).]

Example 3

Are (\vec{a}=(2,−1,3)) and (\vec{b}=(1,5,1)) perpendicular?

Take the dot product:

[\vec{a} \cdot \vec{b} = (2)(1) + (−1)(5) + (3)(1) = 2 - 5 + 3 = 0.]

Example 4

Find the angle between (\vec{a}=(1,0,0)) and (\vec{b}=(1,1,0)).

First the dot product:

[\vec{a} \cdot \vec{b} = (1)(1) + 0 + 0 = 1.]

Final answer: (\theta = 45°).

Example 5

Find the area of the parallelogram spanned by (\vec{a}=(3,0,0)) and (\vec{b}=(0,4,0)).

The area is the magnitude of the cross product:

[\vec{a} \times \vec{b} = (0⋅0−0⋅4, 0⋅0−3⋅0, 3⋅4−0⋅0)=(0,0,12).]

Final answer: area = 12 square units.

Example 6

Find the scalar triple product (\vec{a} \cdot (\vec{b} \times \vec{c})) for (\vec{a} = (1,0,0)), (\vec{b} = (0,1,0)), (\vec{c} = (0,0,1)).

[\vec{b} \times \vec{c} = (1,0,0).]

Final answer: 1 — the volume of the unit cube these three vectors span.

Key Takeaways

• The product of vectors splits into two operations: the dot product (returns a scalar) and the cross product (returns a vector).
• Use the dot product for angles, work, and projections; use the cross product for torque, area, and perpendicular directions.
• The cross product is not commutative — swapping the order flips the sign — while the dot product is.