Vector Subtraction

The Definition of Subtraction

Properties of Subtraction

The Geometric Meaning

Vector subtraction can be understood as the inverse operation of vector addition, and its geometric meaning can be grasped in two ways.

The Relation between Vector Subtraction and the Opposite Vector

The vector subtraction $\vec{a} - \vec{b}$ can be turned into the vector addition $\vec{a} + (-\vec{b})$, that is, negate the vector $\vec{b}$ and add it to $\vec{a}$.

The Position Vector Representation

In a coordinate system, the position vector from the point $A$ to the point $B$ is $\vec{AB} = \vec{b} - \vec{a}$, where $\vec{a}$ and $\vec{b}$ are the position vectors of the points $A$ and $B$ respectively. This means that starting from $A$ and moving in the direction of $\vec{b} - \vec{a}$ brings you to $B$.

Worked Examples

Position Vectors

In a coordinate system, the position vector from the point $A$ to the point $B$ is $\vec{AB} = \vec{b} - \vec{a}$, where $\vec{a}$ and $\vec{b}$ are the position vectors of the points $A$ and $B$ respectively.

Relative Displacement

The displacement vector needed to move an object from the position $\vec{a}$ to the position $\vec{b}$ is: $\vec{b} - \vec{a}$.

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Exercises

Exercise 1

Given $\vec{a} = (2, 3, 1)$ and $\vec{b} = (1, -1, 2)$, find $\vec{a} - \vec{b}$.

Exercise 2

Given the three points $A(1, 2, 3)$, $B(2, 3, 4)$ and $C(3, 4, 5)$, find the vectors $\vec{AB}$ and $\vec{AC}$.

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Summary

Symbols Used in This Article