Scalar Multiplication If $\mathbf{v}$ is a nonzero vector in 2 -space or 3 -space, and if $k$ is a nonzero scalar, then we define the scalar product of $\mathbf{v}$ by $\mathbf{k}$ to be the vector whose length is $|k|$ times the length of $\mathbf{v}$ and whose direction is the same as that of $\mathbf{v}$ if $k$ is positive and opposite to that of $\mathbf{v}$ if $k$ is negative. If $k=0$ or $\mathbf{v}=\mathbf{0}$, then we define $k \mathbf{v}$ to be $\mathbf{0}$.

The diagram below shows the geometric relationship between a vector **v** and some of its scalar multiples. In particular, observe that **(-1)v** has the same length as $\mathbf{v}$ but is oppositely directed; therefore,

$$

(-1) \mathbf{v}=-\mathbf{v}

$$