Operations with vectors

We can carry out basic operations with vectors, such as addition and multiplication.

Adding vectors

To add two vectors, draw both from the origin of the coordinate system, complete the parallelogram, and the diagonal starting at the origin is the resultant vector. There is of course an illustration:

Analytically, the sum of vectors is the sum of the corresponding coordinates. So if you have two vectors $\overrightarrow{\mathbf{u}}=(u_1,u_2)$ and $\overrightarrow{\mathbf{v}}=(v_1,v_2)$, then the sum $\overrightarrow{\mathbf{u}}+\overrightarrow{\mathbf{v}}$ is

$$\overrightarrow{\mathbf{u}}+\overrightarrow{\mathbf{v}}=(u_1+v_1,u_2+v_2)$$

For the vectors in the picture, $\overrightarrow{\mathbf{u}}=(2,4)$ and $\overrightarrow{\mathbf{v}}=(4,1)$. The sum then looks like this: $\overrightarrow{\mathbf{u}}+\overrightarrow{\mathbf{v}}=(2+4,4+1)=(6,5)$. These coordinates match the point D.

Subtracting vectors is the same as adding the negative vector. Analytically:

$$\overrightarrow{\mathbf{u}}-\overrightarrow{\mathbf{v}}=(u_1-v_1,u_2-v_2)$$

If we add vectors that lie on the same line and point the same way, we simply stretch the resultant vector. If they point in opposite directions, we subtract their magnitudes. This is easier to see in a picture:

Vector addition is commutative and associative. There is a vector $\overrightarrow{\mathbf{0}}$, called the zero vector, for which $\overrightarrow{\mathbf{u}}+\overrightarrow{\mathbf{0}}=\overrightarrow{\mathbf{u}}$, just as with numbers. For every vector $\overrightarrow{\mathbf{u}}$ there is a negative vector $-\overrightarrow{\mathbf{u}}$ for which $\overrightarrow{\mathbf{u}}+(-\overrightarrow{\mathbf{u}})=\overrightarrow{\mathbf{0}}$.

Multiplying a vector by a number

If you multiply a vector by a real number k, you just multiply both of its coordinates by k. Geometrically this “stretches” or “shrinks” the vector, or flips it round if k is negative.

The picture shows that when we multiply the vector u by the number k:

• If the absolute value of k is less than one, the vector gets shorter.
• If the absolute value of k is greater than one, the vector gets longer.
• If k is negative, the vector points the opposite way.

Linear combinations of vectors

In linear algebra we often use linear combinations of vectors. Given vectors ${\overrightarrow{\mathbf{u}}}_1,{\overrightarrow{\mathbf{u}}}_2,{\overrightarrow{\mathbf{u}}}_3,\dots$, a linear combination of these vectors is

• one of the vectors ${\overrightarrow{\mathbf{u}}}_n$ multiplied by a scalar k,
• the sum of any two or more of the vectors,
• a combination of the previous two: we can add scalar multiples k_n of any of the vectors ${\overrightarrow{\mathbf{u}}}_n$.

Given vectors ${\overrightarrow{\mathbf{u}}}_1,{\overrightarrow{\mathbf{u}}}_2,\dots ,{\overrightarrow{\mathbf{u}}}_n$, the vector $\overrightarrow{\mathbf{v}}$ is a linear combination of the vectors ${\overrightarrow{\mathbf{u}}}_n$ if

$$\overrightarrow{\mathbf{v}}=c_1\cdot {\overrightarrow{\mathbf{u}}}_1+c_2\cdot {\overrightarrow{\mathbf{u}}}_2+\dots +c_n\cdot {\overrightarrow{\mathbf{u}}}_n;c_i\in \mathbb{R}$$

Notice that the coefficients c_i are real numbers, so we can choose zero for any of them, which makes that vector drop out completely. For example, take the vectors ${\overrightarrow{\mathbf{u}}}_1=(1,3),{\overrightarrow{\mathbf{u}}}_2=(0,4),{\overrightarrow{\mathbf{u}}}_3=(7,2)$. Here are some of the possible linear combinations:

$$\begin{array}{rcl}(8,9)& =& 1\cdot (1,3)+1\cdot (0,4)+1\cdot (7,2)\\ (22,17)& =& 1\cdot (1,3)+2(0,4)+3(7,2)\\ (68,38)& =& -2(1,3)+6(0,4)+10(7,2)\\ (-\frac{1}{2},-\frac{43}{2})& =& -\frac{1}{2}(1,3)-5(0,4)+0(7,2)\end{array}$$