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Add and subtract vectors

Vector addition and subtraction are understood as combining directed quantities: geometrically, vectors are placed head-to-tail or represented by the parallelogram rule, while in component form corresponding horizontal, vertical, and spatial components are added or subtracted. The result is interpreted as a resultant vector, with subtraction understood as adding the opposite vector; learners distinguish a vector’s magnitude from its direction and recognize that addition is commutative, whereas subtraction is not. The scope is limited to ordinary Euclidean vectors in two or three dimensions, not abstract vector spaces or advanced vector operations.

Detailed Explanation: Add and subtract vectors

Vectors can be added or subtracted by working with their corresponding components.

  • Add horizontal components together and vertical components together.
  • For subtraction, subtract each corresponding component. This is the same as adding the opposite vector:
a−b=a+(−b)\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b})

Worked example

Let

a=(34),b=(−12).\mathbf{a}=\begin{pmatrix}3\\4\end{pmatrix}, \qquad \mathbf{b}=\begin{pmatrix}-1\\2\end{pmatrix}.

Find a+b\mathbf{a}+\mathbf{b} and a−b\mathbf{a}-\mathbf{b}.

1. Add the vectors

Add the corresponding components:

a+b=(34)+(−12)=(3+(−1)4+2)=(26).\mathbf{a}+\mathbf{b} = \begin{pmatrix}3\\4\end{pmatrix} + \begin{pmatrix}-1\\2\end{pmatrix} = \begin{pmatrix}3+(-1)\\4+2\end{pmatrix} = \begin{pmatrix}2\\6\end{pmatrix}.

So, the resultant vector is

a+b=(26).\boxed{\mathbf{a}+\mathbf{b}=\begin{pmatrix}2\\6\end{pmatrix}}.

Geometrically, this means place b\mathbf{b} at the head of a\mathbf{a}. The resultant goes from the tail of a\mathbf{a} to the head of the moved vector b\mathbf{b}.

2. Subtract the vectors

Subtract the corresponding components:

a−b=(34)−(−12)=(3−(−1)4−2)=(42).\mathbf{a}-\mathbf{b} = \begin{pmatrix}3\\4\end{pmatrix} - \begin{pmatrix}-1\\2\end{pmatrix} = \begin{pmatrix}3-(-1)\\4-2\end{pmatrix} = \begin{pmatrix}4\\2\end{pmatrix}.

Therefore,

a−b=(42).\boxed{\mathbf{a}-\mathbf{b}=\begin{pmatrix}4\\2\end{pmatrix}}.

Equivalently, reverse the direction of b\mathbf{b}:

−b=(1−2),-\mathbf{b}=\begin{pmatrix}1\\-2\end{pmatrix},

then add:

a−b=a+(−b)=(34)+(1−2)=(42).\mathbf{a}-\mathbf{b} = \mathbf{a}+(-\mathbf{b}) = \begin{pmatrix}3\\4\end{pmatrix} + \begin{pmatrix}1\\-2\end{pmatrix} = \begin{pmatrix}4\\2\end{pmatrix}.

The vector (42)\begin{pmatrix}4\\2\end{pmatrix} describes both a direction and a magnitude. Its magnitude is

∣a−b∣=42+22=20=25.\left \vert \mathbf{a}-\mathbf{b}\right \vert = \sqrt{4^2+2^2} = \sqrt{20} = 2\sqrt{5}.

Remember that addition is commutative, so a+b=b+a\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}. Subtraction is not commutative: usually a−b≠b−a\mathbf{a}-\mathbf{b}\ne\mathbf{b}-\mathbf{a}.

Learn by doing: Add and subtract vectors

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    Vectors - Addition - Vectors to Sum

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