3.2 Norm, Dot Product, and Distance in Rn

Exercise

1.a

Find the norm of $\mathbf{v}$, and a unit vector that is oppositely directed to $\mathbf{v}$.

Let $\mathbf{v}=(2,2,2)$

Norm of $\mathbf{v}$ is

$$\begin{array}{rl}||\mathbf{v}||& =\sqrt{2^2+2^2+2^2}\\ & =\sqrt{12}\\ & =2\sqrt{3}\end{array}$$

The unit vector of $\mathbf{v}$ is given by

$$\begin{array}{rl}\mathbf{u}& =\frac{1}{||\mathbf{v}||}\mathbf{v}\\ & =\frac{1}{2\sqrt{3}}(2,2,2)\\ & =\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)\end{array}$$

The unit vector result can be confirmed using the property of unit vector,

$$\begin{array}{rl}||\mathbf{u}||& =1\\ \left|\left|\left(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right)\right|\right|& =1\\ \sqrt{{\left(\frac{1}{\sqrt{3}}\right)}^2+{\left(\frac{1}{\sqrt{3}}\right)}^2+{\left(\frac{1}{\sqrt{3}}\right)}^2}& =1\\ \sqrt{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}& =1\\ \sqrt{1}& =1\\ 1& =1\end{array}$$

1.b

Find the norm of $\mathbf{v}$, and a unit vector that is oppositely directed to $\mathbf{v}$.

Let $\mathbf{v}=(1,0,2,1,3)$

$$\begin{array}{rl}||\mathbf{v}||& =\sqrt{1^2+0^2+2^2+1^2+3^2}=\sqrt{15}\\ \mathbf{u}& =\frac{1}{||\mathbf{v}||}\mathbf{v}=\frac{1}{\sqrt{15}}(1,0,2,1,3)\end{array}$$

2.a

Find the norm of $\mathbf{v}$, and a unit vector that is oppositely directed to $\mathbf{v}$.

Let $\mathbf{v}=(1,-1,2)$

$$\begin{array}{rl}||\mathbf{v}||& =\sqrt{1^2+(-1{)}^2+2^2}=\sqrt{6}=\frac{1}{2}\sqrt{2}\\ \mathbf{u}& =\frac{1}{||\mathbf{v}||}\mathbf{v}=\frac{1}{\sqrt{6}}(1,-1,2)=\left(\frac{1}{2}\sqrt{2},-\frac{1}{2}\sqrt{2},\sqrt{2}\right)\end{array}$$

2.b

Find the norm of $\mathbf{v}$, and a unit vector that is oppositely directed to $\mathbf{v}$.

Let $\mathbf{v}=(-2,3,3,-1)$

$$\begin{array}{rl}||\mathbf{v}||& =\sqrt{(-2{)}^2+3^2+3^2+(-1{)}^2}=\sqrt{23}\\ \mathbf{u}& =\frac{1}{||\mathbf{v}||}\mathbf{v}=\frac{1}{\sqrt{23}}(-2,3,3,-1)\end{array}$$

3.a

Evaluate the given expression with $\mathbf{u}=(2,-2,3),\mathbf{v}=(1-3,4)$, and $\mathbf{w}=(3,6,-4)$

$$\begin{array}{rl}||\mathbf{u}+\mathbf{v}||& =||(2,-2,3)+(1,-3,4)||\\ & =||(3,-5,7)||\\ & =\sqrt{3^2+(-5{)}^2+7^2}\\ & =\sqrt{83}\end{array}$$

3.b

Evaluate the given expression with $\mathbf{u}=(2,-2,3),\mathbf{v}=(1-3,4)$, and $\mathbf{w}=(3,6,-4)$

$$\begin{array}{rl}||\mathbf{u}||+||\mathbf{v}||& =||(2,-2,3)||+||(1,-3,4)||\\ & =\sqrt{2^2+(-2{)}^2+3^2}+\sqrt{1^2+(-3{)}^2+4^2}\\ & =\sqrt{17}+\sqrt{26}\end{array}$$

3.c

Evaluate the given expression with $\mathbf{u}=(2,-2,3),\mathbf{v}=(1,-3,4)$, and $\mathbf{w}=(3,6,-4)$

$$\begin{array}{rl}||-2\mathbf{u}+2\mathbf{v}||& =||-2(2,-2,3)+2(1,-3,4)||\\ & =||(-4,4,6)+(2,-6,8)||\\ & =||(-2,-2,14)||\\ & =\sqrt{(-2{)}^2+(-2{)}^2+14^2}\\ & =\sqrt{204}\end{array}$$