Linear Algebra ›Vectors ›Addition and Subtraction

Adding and Subtracting Vectors

Addition and subtraction are both defined for vectors. They actually work rather intuitively.

Addition

Addition is defined as follows;

Presume we want to sum two vectors v⃗\vec{v} and w⃗\vec{w} where

v⃗=[v1v2],w⃗=[w1w2].\vec{v} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}, \vec{w} = \begin{bmatrix} w_1 \\ w_2 \end{bmatrix}.

Then, their sum is

v⃗+w⃗=[v1v2]+[w1w2]=[v1+w1v2+w2]\vec{v} + \vec{w} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} + \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} = \begin{bmatrix} v_1 + w_1 \\ v_2 + w_2 \end{bmatrix}

Subtraction

Subtraction follows the same logic, where for the two vectors v⃗\vec{v} and w⃗\vec{w} defined above

v⃗−w⃗=[v1v2]−[w1w2]=[v1−w1v2−w2]\vec{v} - \vec{w} = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} - \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} = \begin{bmatrix} v_1 - w_1 \\ v_2 - w_2 \end{bmatrix}

Commutativity

From the definition above, it is evident that the same commutativity that holds for addition of real numbers R\mathbb{R}, also holds for vector addition:

v⃗+w⃗=[v1v2]+[w1w2]=[v1+w1v2+w2]=[w1+v1w2+v2]=[w1w2]+[v1v2]=w⃗+v⃗\begin{align*} \vec{v} + \vec{w} &= \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} + \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} \\ &= \begin{bmatrix} v_1 + w_1 \\ v_2 + w_2 \end{bmatrix} \\ &= \begin{bmatrix} w_1 + v_1 \\ w_2 + v_2 \end{bmatrix} \\ &= \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} + \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} \\ &= \vec{w} + \vec{v} \end{align*}

Graphical Representation

If you’re adding a vector v⃗\vec{v} to a vector w⃗\vec{w}, the resulting vector v⃗+w⃗\vec{v} + \vec{w} will be the result of moving w⃗\vec{w} in such a way, such that w⃗\vec{w}‘s starting point will be at v⃗\vec{v}‘s end, as shown in the following figure:

In the example depicted above, the vectors are the following:

v⃗=[42],w⃗=[46],v⃗+w⃗=[88]\vec{v} = \begin{bmatrix} 4 \\ 2 \end{bmatrix}, \vec{w} = \begin{bmatrix} 4 \\ 6 \end{bmatrix}, \vec{v} + \vec{w} = \begin{bmatrix} 8 \\ 8 \end{bmatrix}