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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 a,b∈Rn\mathbf{a}, \mathbf{b} \in \mathbb{R}^n and a scalar c∈Rc \in \mathbb{R}, vector multiplication encompasses: (1) scalar multiplication cac\mathbf{a}, which scales each component; (2) the dot product a⋅b=∑i=1naibi\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i, which returns a scalar; and (3) the cross product a×b\mathbf{a} \times \mathbf{b} (defined for R3\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 a=⟨1,2,3⟩\mathbf{a} = \langle 1, 2, 3 \rangle and b=⟨4,−1,2⟩\mathbf{b} = \langle 4, -1, 2 \rangle. Compute (a) 3a3\mathbf{a}, (b) a⋅b\mathbf{a} \cdot \mathbf{b}, and (c) a×b\mathbf{a} \times \mathbf{b}.
Scalar multiplication: Multiply each component of a\mathbf{a} by 3.
3a=⟨3(1),  3(2),  3(3)⟩=⟨3,6,9⟩3\mathbf{a} = \langle 3(1),\; 3(2),\; 3(3) \rangle = \langle 3, 6, 9 \rangle
Dot product: Multiply corresponding components and add.
a⋅b=(1)(4)+(2)(−1)+(3)(2)=4−2+6=8\mathbf{a} \cdot \mathbf{b} = (1)(4) + (2)(-1) + (3)(2) = 4 - 2 + 6 = 8
Cross product: Use the determinant formula with i^,j^,k^\hat{i}, \hat{j}, \hat{k}.
a×b=⟨(2)(2)−(3)(−1),  (3)(4)−(1)(2),  (1)(−1)−(2)(4)⟩=⟨7,10,−9⟩\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: 3a=⟨3,6,9⟩3\mathbf{a} = \langle 3, 6, 9 \rangle, a⋅b=8\mathbf{a} \cdot \mathbf{b} = 8, a×b=⟨7,10,−9⟩\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.

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