5. Vectors

c. Scalar Multiplication

3. Properties

We here list the algebraic properties of scalar multiplication and its relation to vector addition and magnitude.

In the following, \(a\) and \(b\) are scalars (real numbers) while \(\vec u\), \(\vec v\), and \(\vec w\) are vectors. Also, \(\vec 0\) is the vector whose components are all zero and \(-\vec v\) denotes the vector whose components are the negatives of those of \(\vec v\).

First, there are \(2\) properties involving just scalar multiplication.

Let \(a\) and \(b\) be arbitrary scalars and \(\vec v\) be an arbitrary vector. Then,

  1. Scalar Multiplication is Associative:   \((ab)\vec v=a(b\vec v)\)  

\[\begin{aligned} (ab)\vec v &=\left\langle abv_1,abv_2\right\rangle \\ &=a\left\langle bv_1,bv_2\right\rangle =a(b\vec v) \end{aligned}\]

  1. \(1\) is the Multiplicative Identity:   \(1\,\vec v=\vec v\)  

\[ 1\,\vec v=\left\langle1v_1,1v_2\right\rangle =\left\langle v_1,v_2\right\rangle=\vec v \]

Second, there are \(6\) properties relating vector addition and scalar multiplication.

Let \(a\) and \(b\) be arbitrary scalars and \(\vec u\) and \(\vec v\) be arbitrary vectors. Then,

  1. Scalar Multiplication Distributes over Ordinary Addition:   \((a+b)\vec v=a\vec v+b\vec v\)  

\[\begin{aligned} (a+b) \vec v &=\left\langle(a+b)v_1,(a+b)v_2\right\rangle \\ &=\left\langle av_1+bv_1,av_2+bv_2\right\rangle \\ &=\left\langle av_1,av_2\right\rangle+\left\langle bv_1,bv_2\right\rangle \\ &=a\vec v+b\vec v \end{aligned}\]

  1. Scalar Multiplication Distributes over Vector Addition:   \(a(\vec u+\vec v)=a\vec u+a\vec v\)  

\[\begin{aligned} a(\vec u+\vec v) &=a\left\langle u_1+v_1,u_2+v_2\right\rangle \\ &=\left\langle a(u_1+v_1),a(u_2+v_2)\right\rangle \\ &=\left\langle au_1+av_1,au_2+av_2\right\rangle \\ &=\left\langle au_1,au_2\right\rangle+\left\langle av_1,av_2\right\rangle \\ &=a\vec u+a\vec v \end{aligned}\]

  1. \((-1)\vec v\) is the Negative of \(\vec v\):   \((-1)\vec v=-\vec v\)  

\[ (-1)\,\vec v=\left\langle-1v_1,-1v_2\right\rangle=-\vec v \]

  1. \(0\) times any Vector is \(\vec 0\):   \(0\,\vec v=\vec0\)  

\[ 0\,\vec v=\left\langle0v_1,0v_2\right\rangle =\left\langle 0,0\right\rangle=\vec0 \]

  1. Any Scalar times \(\vec0\) is \(\vec0\):   \(a\vec0=\vec0\)  

\[ a\vec0=\left\langle a0,a0\right\rangle =\left\langle0,0\right\rangle=\vec0 \]

  1. Zero Product: If \(a\vec v=\vec0\), then either \(a=0\) or \(\vec v=\vec0\):  

\[\begin{aligned} &&a\vec v&=\vec0 \\ &\implies&\left\langle av_1,av_2\right\rangle &=\left\langle0,0\right\rangle \\ &\implies&av_1=0, &\quad av_2=0 \\ \end{aligned}\] So either \(a=0\) or \(v_1=v_2=0\). In the latter case, \(\vec v=\vec0\).

Finally, we have the relation between scalar multiplication and magnitude.

Let \(a\) be an arbitrary scalar and \(\vec v\) be an arbitrary vector. Then,

  1. Scalar Multiplication stretches or shrinks a vector:   \(|a\,\vec v|=|a|\,|\vec v|\)  

\[\begin{aligned} |a\vec v| &=\sqrt{\left(a\,v_1\right)^2+\left(av_2\right)^2} =\sqrt{a^2\left({v_1}^2+{v_2}^2\right)} \\ &=\sqrt{a^2}\sqrt{ {v_1}^2+{v_2}^2} =|a|\,|\vec v| \end{aligned}\]

Optional Notice that properties 1-8 above were proved by writing out the components of each vector and checking the formula. However, properties 5-8 can also be proved using only Vector Addition Properties 1-4 and Scalar Multiplication Properties 1-4 without writing out any components.
These proofs will be assigned in the exercises.

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