Equality Two vectors $\vec{u}$ and $\vec{v}$ in a geometric plane are equal to each other, $\vec{u}=\vec{v}$, if they have the same length and direction.

Opposite The opposite of a vector $\vec{v}$ is defined as the vector with the same length but having the opposite direction. We denote the opposite of vector $\vec{v}$ as $-\vec{v}$.

Scalar multiplication The product of a number $\lambda$ with a vector $\vec{v}$ is defined as the vector obtained from$\vec{v}$ by scaling it relative to its initial point with a scale factor $|\lambda|$, provided that we first reverse the direction of the vector $\vec{v}$ if $\lambda \lt 0$. We write this product as $\lambda \cdot\vec{v}$.

So, $\lambda \cdot\vec{v}$ is the vector whose

• length is $\left|\lambda\right|$ times the length of $\vec{v}$ , and
• direction is equal to that of $\vec{v}$ if $\lambda\gt 0$ and equal to that of $-\vec{v}$ if $\lambda \lt 0$.

We call $\lambda \cdot\vec{v}$ a scalar multiple of $\vec{v}$ or simply a multiple of $\vec{v}$. The (scaling) number $\lambda$ with which we scale the vector is called a scalar. Usually we write

• $\vec{v}$ instead of $1\cdot\vec{v}$;
• $-\vec{v}$ instead of $(-1)\cdot\vec{v}$;
• $2\,\vec{v}$ instead of $2\cdot\vec{v}$, and so on;
• $-2\cdot \vec{v}$ and $-2\,\vec{v}$ instead of $(-2)\cdot\vec{v}$, and so on.

All scalar multiples of a vector $\vec{v}$ form together the span of $\vec{v}$, denoted as $\langle\vec{v}\rangle$.

Sum and difference of vectors Two position vectors can be added and the result is again a position vector, namely the one obtained through the construction of a parallelogram:

Move $\vec{u}$ and/or $\vec{v}$ to observe how the sum of two position vectors can be determined.

The sum of two vectors $\vec{u}$ and $\vec{v}$ can also be obtained by laying $\vec{u}$ and $\vec{v}$ head-to-tail, as shown in the figure below:

Move $\vec{u}$ and/or $\vec{v}$ to see how adding vectors goes in this way. If one of the two vectors is a scalar multiple of the other, you can only use the second construction.

What remains is the definition of the sum of the zero vector with a vector $\vec{v}$ as $\vec{v}+\vec{0}=\vec{0}+\vec{v}=\vec{v}$.

Instead of $\vec{u}+(-\vec{v})$ we usually write $\vec{u}-\vec{v}$. This expression is called the difference of $\vec{u}$ and $\vec{v}$; we subtract $\vec{v}$ from $\vec{u}$.