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1.4 The scalar product

The next important operation on vectors is the scalar product, sometimes called the dot product of two vectors.

Definition 1.7

The scalar product of two vectors $(x_{1}\;\;\ldots\;\;x_{n})$ , $(y_{1}\;\;\ldots\;\;y_{n})$ in ${\mathbb{R}}^{n}$ is the real number:

(x_{1}\;\;\ldots\;\;x_{n})\cdot(y_{1}\;\;\ldots\;\;y_{n})=x_{1}y_{1}+\ldots+x_% {n}y_{n}

Note that ${\bf u}\cdot{\bf v}$ is a scalar (as the name suggests), not a vector!

Example 1.8

Compute ${\bf u}\cdot{\bf v}$ for the following pairs of vectors: (i) ${\bf u}=(1\;\;3),{\bf v}=(-1\;\;1)$ , (ii) ${\bf u}=(0\;\;2),{\bf v}=(1\;\;0)$ , (iii) ${\bf u}=(4\;\;-3\;\;\;1),{\bf v}=(2\;\;5\;\;\;-1)$ .

Note that we can only calculate the scalar product of two vectors of the same size! So for example, $(2\;\;1)\cdot(1\;\;0\;\;-1)$ makes no sense.

Theorem 1.9

The scalar product has the following properties:

(i) ${\bf u}\cdot{\bf v}={\bf v}\cdot{\bf u}$ for any ${\bf u},{\bf v}\in{\mathbb{R}}^{n}$ ,

(ii) ${\bf u}\cdot({\bf v}+{\bf w})={\bf u}\cdot{\bf v}+{\bf u}\cdot{\bf w}$ for any ${\bf u}$ , ${\bf v}$ , ${\bf w}\in{\mathbb{R}}^{n}$ ,

(iii) ${\bf u}\cdot(\lambda{\bf v})=\lambda({\bf u}\cdot{\bf v})=(\lambda{\bf u})% \cdot{\bf v}$ for any ${\bf u},{\bf v}\in{\mathbb{R}}^{n}$ , $\lambda\in{\mathbb{R}}$ .

(iv) ${\bf u}\cdot{\bf u}=|{\bf u}|^{2}$ for any ${\bf u}\in{\mathbb{R}}^{n}$ .

Proof

Suppose ${\bf u}=(u_{1}\;\;\ldots\;\;u_{n})$ , ${\bf v}=(v_{1}\;\;\ldots\;\;v_{n})$ , ${\bf w}=(w_{1}\;\;\ldots\;\;w_{n})$ .

(i) ${\bf u}\cdot{\bf v}=u_{1}v_{1}+\ldots+u_{n}v_{n}=v_{1}u_{1}+\ldots+v_{n}u_{n}=% {\bf v}\cdot{\bf u}$ .

(ii) ${\bf u}\cdot({\bf v}+{\bf w})=u_{1}(v_{1}+w_{1})+\ldots+u_{n}(v_{n}+w_{n})=u_{% 1}v_{1}+\ldots_{+}u_{n}v_{n}+u_{1}w_{1}+\ldots+u_{n}w_{n}={\bf u}\cdot{\bf v}+% {\bf u}\cdot{\bf w}$ .

(iii) ${\bf u}\cdot(\lambda{\bf v})=u_{1}(\lambda v_{1})+\ldots+u_{n}(\lambda v_{n})=% \lambda u_{1}v_{1}+\ldots+\lambda u_{n}v_{n}=\lambda{\bf u}\cdot{\bf v}$ .

(iv) ${\bf u}\cdot{\bf u}=u_{1}^{2}+\ldots+u_{n}^{2}$ . By definition, $|{\bf u}|=\sqrt{u_{1}^{2}+\ldots+u_{n}^{2}}$ . $\square$

The scalar product and orthogonal vectors

The scalar product is extremely useful for calculating angles between vectors. A special case is when two vectors are at right-angles to each other. For example, consider two vectors going along the two coordinate axes in ${\mathbb{R}}^{2}$ : the unit vector along the $x$ -axis is $(1\;\;0)$ , while the unit vector along the $y$ -axis is $(0\;\;1)$ . We note that the scalar product of these two vectors is zero: $(1\;\;0)\cdot(0\;\;1)=1\cdot 0+0\cdot 1=0$ . Also, the $x$ and $y$ coordinate axes are at right angles to each other. More generally:

Example 1.10

If ${\bf u}$ is a unit vector at an angle $\theta$ to the $x$ -axis and ${\bf v}$ is a unit vector at an angle $(\theta+\pi/2)$ to the $x$ -axis (i.e. at right-angles to ${\bf u}$ ) then:

{\bf u}=(\cos\theta\;\;\sin\theta),\;\;{\bf v}=(\cos(\theta+\frac{\pi}{2})\;\;% \sin(\theta+\frac{\pi}{2}))=(-\sin\theta\;\;\cos\theta)

So in particular, ${\bf u}\cdot{\bf v}=-\cos\theta\sin\theta+\sin\theta\cos\theta=0$ .

This is an incredibly important property! It is also true in ${\mathbb{R}}^{n}$ .

Theorem 1.11

Let ${\bf u}$ and ${\bf v}$ be two vectors in ${\mathbb{R}}^{n}$ , and let $\theta$ , where $0\leq\theta\leq\pi$ , be the angle between ${\bf u}$ and ${\bf v}$ . Then

{\bf u}\cdot{\bf v}=|{\bf u}||{\bf v}|\cos\theta

Proof

From the law of cosines from trigonometry it follows that

|{\bf v}-{\bf u}|^{2}=|{\bf u}|^{2}+|{\bf v}|^{2}-2|{\bf u}||{\bf v}|\cos\theta.

Since $|{\bf v}-{\bf u}|^{2}=({\bf v}-{\bf u})\cdot({\bf v}-{\bf u})$ and $|{\bf u}|^{2}={\bf u}\cdot{\bf u}$ and $|{\bf v}|^{2}={\bf v}\cdot{\bf v}$ , we can rewrite the above equation as

({\bf v}-{\bf u})\cdot({\bf v}-{\bf u})={\bf u}\cdot{\bf u}+{\bf v}\cdot{\bf v% }-2|{\bf u}||{\bf v}|\cos\theta.

Now

$\displaystyle({\bf v}-{\bf u})\cdot({\bf v}-{\bf u})$	$\displaystyle=$	$\displaystyle{\bf v}\cdot({\bf v}-{\bf u})-{\bf u}\cdot({\bf v}-{\bf u})$
	$\displaystyle=$	$\displaystyle{\bf v}\cdot{\bf v}-{\bf v}\cdot{\bf u}-{\bf u}\cdot{\bf v}+{\bf u% }\cdot{\bf u}$
	$\displaystyle=$	$\displaystyle{\bf u}\cdot{\bf u}+{\bf v}\cdot{\bf v}-2{\bf u}\cdot{\bf v}.$

Thus,

{\bf u}\cdot{\bf u}+{\bf v}\cdot{\bf v}-2{\bf u}\cdot{\bf v}={\bf u}\cdot{\bf u% }+{\bf v}\cdot{\bf v}-2|{\bf u}||{\bf v}|\cos\theta.

This gives the result. $\square$

Corollary 1.12

Two non-zero vectors ${\bf u},{\bf v}\in{\mathbb{R}}^{n}$ are orthogonal if and only if ${\bf u}\cdot{\bf v}=0$ .

Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality - which is a consequence of Thm. 1.11 - is a fundamental result which is often useful in problems of geometric nature.

Theorem 1.13 (Cauchy-Schwarz)

Let ${\bf u},{\bf v}$ be vectors in ${\mathbb{R}}^{n}$ . Then $|{\bf u}\cdot{\bf v}|\leq|{\bf u}|\,|{\bf v}|$ .

Proof

If ${\bf u}$ is not a scalar multiple of ${\bf v}$ , then $|\cos\theta|<1$ and so the inequality holds. In fact, if ${\bf u}$ and ${\bf v}$ are both non-zero, then strict inequality holds in this case. When ${\bf u}$ is a scalar multiple of ${\bf v}$ , then $\theta$ equals zero or $\pi$ and $|\cos\theta|=1$ , so equality holds in this case. $\square$

Let us look at the two extremes that can occur here, which are ${\bf u}\cdot{\bf v}=|{\bf u}|\,|{\bf v}|$ and ${\bf u}\cdot{\bf v}=-|{\bf u}|\,|{\bf v}|$ .

Max. value: ${\bf u}\cdot{\bf v}=|{\bf u}|\,|{\bf v}|$ when ${\bf u}$ and ${\bf v}$ are parallel, and pointing in the same direction,

Min. value: ${\bf u}\cdot{\bf v}=-|{\bf u}|\,|{\bf v}|$ when ${\bf u}$ and ${\bf v}$ are parallel, but pointing in opposite directions.

Example 1.14

Find the greatest and least values of $2x+3y-z$ on the sphere $x^{2}+y^{2}+z^{2}=1$ , and find the values of $x, y, z$ for which these values occur.

Let ${\bf u}=(x\;\;y\;\;z)$ . The condition $x^{2}+y^{2}+z^{2}=1$ can be stated in vector form as: $|{\bf u}|=1$ . The function $2x+3y-z$ is just ${\bf u}\cdot(2\;\;3\;\;-1)$ . But ${\bf u}\cdot(2\;\;3\;\;-1)\leq|{\bf u}|\,|(2\;\;3\;\;-1)|=\sqrt{4+9+1}=\sqrt{14}$ , so the greatest value of $2x+3y-z$ is $\sqrt{14}$ . It occurs when ${\bf u}$ is parallel to $(2\;\;3\;\;-1)$ and pointing in the same direction, so $(x\;\;y\;\;z)=\frac{1}{\sqrt{14}}(2\;\;3\;\;-1)$ .

Similarly, the least value of $2x+3y-z$ is $-\sqrt{14}$ , and it occurs when $(x\;\;y\;\;z)=-\frac{1}{\sqrt{14}}(2\;\;3\;\;-1)$ .

Example 1.15

Find the greatest and least values of $3x-y+z+1$ on the sphere $x^{2}+y^{2}+z^{2}=11$ .

Using scalar products to calculate angles in ${\mathbb{R}}^{n}$

Note that it follows from Thm. 1.11 that the angle $\theta$ , where $0\leq\theta\leq\pi$ , between the non-zero vectors ${\bf u}$ and ${\bf v}$ is given by

\theta=\cos^{-1}\Big{(}\frac{{\bf u}\cdot{\bf v}}{|{\bf u}|\,|{\bf v}|}\Big{)}.

Example 1.16

Find the angle $\theta$ between the vectors ${\bf u}=(3\;\;1)$ and ${\bf v}=(1\;\;2)$ .

Solution: ${\bf u}\cdot{\bf v}=3+2=5$ , while $|{\bf u}|^{2}=3^{2}+1^{2}=10$ , $|{\bf v}|^{2}=1^{2}+2^{2}=5$ , hence $|{\bf u}|\,|{\bf v}|=5\sqrt{2}$ . It follows that $\cos\theta=\frac{5}{5\sqrt{2}}=\frac{1}{\sqrt{2}}$ . Hence $\theta=\frac{\pi}{4}$ .

More commonly you may only be asked to determine the cosine of the angle between ${\bf u}$ and ${\bf v}$ .

Example 1.17

Find $\cos\theta$ , where $\theta$ is the angle between ${\bf u}=(1\;\;2\;\;-1)$ and ${\bf v}=(-1\;\;1\;\;-3)$ .