Chapter 1, Problem 1.1P

To determine

The value of $\text{b}+\text{c, 5b}+2\text{c, b}\cdot \text{c,and b}\times \text{c}$.

Answer to Problem 1.1P

The value of $\text{b}+\text{c}$ is $\underset{\_}{2\widehat{x}+\widehat{y}+\widehat{z}}$, $5\text{b}+2\text{c}$ is $\underset{\_}{7\widehat{x}+5\widehat{y}+2\widehat{z}}$ , $\text{b}\cdot \text{c}$ is 1 and $\text{b}\times \text{c}$ is $\underset{\_}{\widehat{\text{x}}-\widehat{\text{y}}-\widehat{\text{z}}}$

Explanation of Solution

The sum of the given vectors is,

    $\begin{array}{c}\text{b}+\text{c}=\left(\widehat{\text{x}}+\widehat{\text{y}}\right)+\left(\widehat{\text{x}}+\widehat{\text{z}}\right)\\ =2\widehat{\text{x}}+\widehat{\text{y}}+\text{z}\end{array}$        (I)

The sum of the given two vectors is $2\widehat{\text{x}}+\widehat{\text{y}}+\text{z}$.

The sum of the given two vectors to find $5\text{b}+2\text{c}$ is,

    $\begin{array}{c}5\text{b}+\text{c}=5\left(\widehat{\text{x}}+\widehat{\text{y}}\right)+2\left(\widehat{\text{x}}+\widehat{\text{z}}\right)\\ =\widehat{\text{x}}\left(5+2\right)+5\widehat{\text{y}}+\text{2}\widehat{\text{z}}\\ \text{=7}\widehat{\text{x}}+5\widehat{y}+2\widehat{z}\end{array}$        (II)

The value of $\text{b}\cdot \text{c}$ is,

    $\begin{array}{c}\text{b}\cdot \text{c}=\left(\widehat{\text{x}}+\widehat{\text{y}}\right)\cdot \left(\widehat{\text{x}}+\widehat{\text{z}}\right)\\ =\widehat{\text{x}}\cdot \widehat{\text{x}}+\widehat{\text{x}}\cdot \widehat{\text{z}}+\widehat{\text{y}}\cdot \widehat{\text{z}}\\ \text{=1}+0+0+0\\ =1\end{array}$        (III)

The vector or cross product of the vectors is,

    $\text{b}\times \text{c}=\det \left(\begin{array}{ccc}\widehat{\text{x}}& \text{y}& \widehat{\text{z}}\\ 1& 1& 0\\ 1& 0& 1\end{array}\right)$

Solve the above dot product,

$\begin{array}{c}\text{b}\times \text{c}=\widehat{\text{x}}\left(1-0\right)-\widehat{\text{y}}\left(1-0\right)+\widehat{\text{z}}\left(0-1\right)\\ =\widehat{\text{x}}-\widehat{\text{y}}-\widehat{\text{z}}\end{array}$        (IV)

Conclusion:

Therefore, The value of $\text{b}+\text{c}$ is $\underset{\_}{2\widehat{x}+\widehat{y}+\widehat{z}}$, $5\text{b}+2\text{c}$ is $\underset{\_}{7\widehat{x}+5\widehat{y}+2\widehat{z}}$ , $\text{b}\cdot \text{c}$ is 1 and $\text{b}\times \text{c}$ is $\underset{\_}{\widehat{\text{x}}-\widehat{\text{y}}-\widehat{\text{z}}}$