5. Vectors

c. Scalar Multiplication

1. Algebraic Definition

The scalar product of a scalar (real number), \(c\), and a vector, \(\vec v=\left\langle v_1,v_2\right\rangle\), is the vector whose components are the components of the original vector each multiplied by the scalar: \[ c\,\vec v=\left\langle c\,v_1,c\,v_2\right\rangle \]

If a particle moves \(2\) units east and \(3\) units north, then the displacement vector to describe this motion is \(\vec u=\left\langle2,3\right\rangle\). If this motion is repeated \(4\) times, then the total motion is \(4\vec u=4\left\langle2,3\right\rangle=\left\langle8,12\right\rangle\). That is \(8\) units east and \(12\) units north.

Recall that we previously defined the direction of a vector \(\vec v=\left\langle v_1,v_2\right\rangle\) to be the unit vector \[ \hat v =\left\langle\dfrac{v_1}{|\vec v|},\dfrac{v_2}{|\vec v|}\right\rangle \] and used the shorthand \[ \hat v=\dfrac{\vec v}{|\vec v|} \] for this vector. We now recognize that this notation is justified because it is the scalar multiplication of \(\vec v\) by the scalar \(\dfrac{1}{|\vec v|}\). Further, this equation can be solved for \(\vec v\) to give:

Every vector is the scalar product of its magnitude and its direction: \[ \vec v=|\vec v|\,\hat v \]