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Section 3.5 Multiplying Vectors by a Number

We have already talked about multiplying a vector by a regular number. For instance, if we multiply a vector \(\vec A = (\text{magnitude } = 10 \text{ cm}, \text{ direction to East}) \) by \(5 \text{,}\) we will get another vector \(\vec B\) that is \(5 \) times as long and in the same direction as \(\vec A\text{.}\) We can write the resulting vector as \(5 \vec A\text{.}\)

\begin{equation*} \vec B = 5 \vec A \end{equation*}

On the other hand, if we multiply by \(-5 \text{,}\) we will get a vector \(\vec C \text{,}\) which is also \(5 \) times as long, but its direction will be opposite that of \(\vec A \text{.}\)

\begin{equation*} \vec C = -5 \vec A = -\vec B \end{equation*}

In the next sections, we will talk about multiplying two arbitrary vectors \(\vec A \) and \(\vec B\text{.}\) There are two different types of multiplication between two vectors, one results in a number and the other results in another vector.