Vector addition and scalar multiplication

Parametric equations

Vectors $\mathbf{u},\mathbf{v}\in {\mathbb{R}}^2$ add by the parallelogram rule. The same sum is the third side of the triangle that goes along $\mathbf{u}$ and then along $\mathbf{v}$. Scalar multiplication $c\mathbf{v}$ changes the length of $\mathbf{v}$. When $c<0$ it also reverses the direction, and when $c=0$ the scaled arrow is a point. These two operations are what a linear combination $a\mathbf{u}+b\mathbf{v}$ is built from.

The dashed sides of the parallelogram run from the tip of $\mathbf{u}$ to $\mathbf{u}+\mathbf{v}$ and from the tip of $\mathbf{v}$ to $\mathbf{u}+\mathbf{v}$. The gold arrow is the sum. The separate arrow is $c\mathbf{v}$.

Interaction

• Scalar c: drag $c$ from $-2$ to $2$ and watch $c\mathbf{v}$ stretch, shrink, and reverse
• Component lines: dashed lines drop the tips of $\mathbf{u}$, $\mathbf{v}$, and $c\mathbf{v}$ onto the two axes
• Drag either endpoint. The sum $\mathbf{u}+\mathbf{v}$ and the parallelogram update together

Related

• Geometric significance of the dot product
• Vector projection
• Linear transformation grid

Further reading

• Euclidean vector (Wikipedia)
• Parallelogram law (Wikipedia)