# Question #00290

Jun 18, 2016

Yes, scalars can be negative.

#### Explanation:

While a scalar is a magnitude without a direction by itself, typically we talk about them in tandem with vectors (they are used to "scale" the vectors via scalar multiplication). A negative scalar has meaning in that it produces a vector with an opposite direction from the vector it is scaling.

While this does not perfectly coincide with the concept of a magnitude representing size (which would be positive), it is convenient to use this notation. It may be helpful to think of the magnitude of a scalar as what scales the vector, and the sign of the scalar as what dictates the vector's direction.

If we are considering the length of a vector, (such as in a normed vector space) we do restrict ourselves to non-negative values. If we only consider a vector's length, and not it's direction, then only the absolute value of a scalar matters. That is to say, given a vector $\vec{v}$ with length $| | \vec{v} | |$ and a scalar $s$, we have

$| | a \vec{v} | | = | a | \cdot | | \vec{v} | |$