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vector subtraction — Definition, Formula & Examples

Vector subtraction is the operation of finding the difference between two vectors by subtracting each corresponding component of the second vector from the first. The result is a new vector that points from the tip of the second vector to the tip of the first.

Given two vectors a=(a1,a2,…,an)\mathbf{a} = (a_1, a_2, \ldots, a_n) and b=(b1,b2,…,bn)\mathbf{b} = (b_1, b_2, \ldots, b_n) in Rn\mathbb{R}^n, their difference is defined as a−b=(a1−b1, a2−b2, …, an−bn)\mathbf{a} - \mathbf{b} = (a_1 - b_1,\, a_2 - b_2,\, \ldots,\, a_n - b_n). Equivalently, vector subtraction is the addition of a\mathbf{a} and the additive inverse −b-\mathbf{b}.

Key Formula

a−b=(a1−b1,  a2−b2,  a3−b3)\mathbf{a} - \mathbf{b} = (a_1 - b_1,\; a_2 - b_2,\; a_3 - b_3)
Where:
  • a\mathbf{a} = The first vector, with components $(a_1, a_2, a_3)$
  • b\mathbf{b} = The vector being subtracted, with components $(b_1, b_2, b_3)$

How It Works

To subtract vector b\mathbf{b} from vector a\mathbf{a}, subtract each component of b\mathbf{b} from the matching component of a\mathbf{a}. In 2D, if a=(a1,a2)\mathbf{a} = (a_1, a_2) and b=(b1,b2)\mathbf{b} = (b_1, b_2), you compute (a1−b1, a2−b2)(a_1 - b_1,\, a_2 - b_2). Geometrically, you can think of it as reversing b\mathbf{b} (flipping its direction) and then adding it to a\mathbf{a} using the triangle or parallelogram rule. The resulting vector a−b\mathbf{a} - \mathbf{b} points from the tip of b\mathbf{b} to the tip of a\mathbf{a} when both vectors start at the origin.

Worked Example

Problem: Subtract vector b = (7, 2, −1) from vector a = (3, 8, 5).
Step 1: Subtract the x-components.
3−7=−43 - 7 = -4
Step 2: Subtract the y-components.
8−2=68 - 2 = 6
Step 3: Subtract the z-components.
5−(−1)=65 - (-1) = 6
Answer: a−b=(−4,  6,  6)\mathbf{a} - \mathbf{b} = (-4,\; 6,\; 6)

Why It Matters

Vector subtraction is essential in physics for finding relative velocity, displacement between two points, and the direction from one object to another. In computer graphics and game development, subtracting position vectors determines the direction a camera or character should face.

Common Mistakes

Mistake: Subtracting in the wrong order, computing b−a\mathbf{b} - \mathbf{a} instead of a−b\mathbf{a} - \mathbf{b}.
Correction: Unlike scalar subtraction, swapping the order reverses the direction of the result. b−a=−(a−b)\mathbf{b} - \mathbf{a} = -(\mathbf{a} - \mathbf{b}), so always check which vector comes first.

Related Terms