5. Vectors

c. Scalar Multiplication

2. Geometric Construction

How are the magnitude and direction of \(c\,\vec v\) related to those of \(\vec v\)? Knowing this will allow us to give a geometric description of scalar multiplication:

First the magnitude: \[\begin{aligned} |c\,\vec v| &=\sqrt{\left(c\,v_1\right)^2+\left(c\,v_2\right)^2} =\sqrt{c^2\left({v_1}^2+{v_2}^2\right)} \\ &=\sqrt{c^2}\sqrt{ {v_1}^2+{v_2}^2} =|c|\,|\vec v| \end{aligned}\] Notice that \(\sqrt{c^2}=|c|\) not \(c\), in case \(c\) is negative. Thus, the vector \(c\,\vec v\) is \(|c|\) times as long as \(\vec v\).

Next the direction. Recall that the direction is its unit vector. So let \(\vec w=c\,\vec v=\left\langle c\,v_1,c\,v_2\right\rangle\). Then \(|\vec w|=|c|\,|\vec v|\). So \[\begin{aligned} \hat w &=\dfrac{\vec w}{|\vec w|} =\left\langle\dfrac{c\,v_1}{|c|\,|\vec v|}, \dfrac{c\,v_2}{|c|\,|\vec v|}\right\rangle \\ &=\dfrac{c}{|c|} \left\langle\dfrac{v_1}{|\vec v|}, \dfrac{v_2}{|\vec v|}\right\rangle =\dfrac{c}{|c|}\hat v \end{aligned}\] Consequently:

• If \(c \gt 0\), so that \(|c|=c\), then \(\hat w=\hat v\). So \(\vec w=c\,\vec v\) has the same direction as \(\vec v\).
• If \(c \lt 0\), so that \(|c|=-c\), then \(\hat w=-\hat v\). So \(\vec w=c\,\vec v\) points opposite to the direction of \(\vec v\).
• If \(c=0\), then \(0\,\vec v=\left\langle0,0\right\rangle=\vec 0\). This is the zero vector which has \(0\) length and does not point anywhere.

We summarize:

Geometric Construction