Vector Multiplication — Definition, Formula & Examples

Vector multiplication refers to the different ways two vectors (or a scalar and a vector) can be combined to produce a new quantity. The three main types are scalar multiplication, the dot product, and the cross product, each yielding a fundamentally different result.

Given vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R}^n$ and a scalar $c \in \mathbb{R}$ , vector multiplication encompasses: (1) scalar multiplication $c\mathbf{a}$ , which scales each component; (2) the dot product $\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i$ , which returns a scalar; and (3) the cross product $\mathbf{a} \times \mathbf{b}$ (defined for $\mathbb{R}^3$ ), which returns a vector orthogonal to both operands.

How It Works

Scalar multiplication stretches or shrinks a vector by a constant factor — multiply every component by the scalar. The dot product multiplies corresponding components and sums the results, giving a single number that measures how much two vectors point in the same direction. The cross product, available only in three dimensions, produces a new vector perpendicular to the original two, with magnitude equal to the area of the parallelogram they span. Choosing the right type depends on what you need: a scaled vector, a scalar measure of alignment, or a perpendicular direction.

Worked Example

Problem: Let

\mathbf{a} = \langle 1, 2, 3 \rangle

and

\mathbf{b} = \langle 4, -1, 2 \rangle

. Compute (a)

3\mathbf{a}

, (b)

\mathbf{a} \cdot \mathbf{b}

, and (c)

\mathbf{a} \times \mathbf{b}

Scalar multiplication: Multiply each component of

\mathbf{a}

by 3.

3\mathbf{a} = \langle 3(1),\; 3(2),\; 3(3) \rangle = \langle 3, 6, 9 \rangle

Dot product: Multiply corresponding components and add.

\mathbf{a} \cdot \mathbf{b} = (1)(4) + (2)(-1) + (3)(2) = 4 - 2 + 6 = 8

Cross product: Use the determinant formula with

\hat{i}, \hat{j}, \hat{k}

\mathbf{a} \times \mathbf{b} = \langle (2)(2) - (3)(-1),\; (3)(4) - (1)(2),\; (1)(-1) - (2)(4) \rangle = \langle 7, 10, -9 \rangle

Answer:

3\mathbf{a} = \langle 3, 6, 9 \rangle

\mathbf{a} \cdot \mathbf{b} = 8

\mathbf{a} \times \mathbf{b} = \langle 7, 10, -9 \rangle

Why It Matters

Physics and engineering rely on vector multiplication constantly: the dot product computes work done by a force, while the cross product gives torque and magnetic force. In computer graphics, these operations determine lighting angles and surface normals for every rendered frame.

Common Mistakes

Mistake: Confusing the dot product and cross product, or expecting both to return the same type of quantity.

Correction: The dot product returns a scalar, while the cross product returns a vector. They answer different questions — alignment versus perpendicular direction — and are not interchangeable.