Chapter 9, Problem 78RE

To determine

To calculate: To use the vectors $v=3i+j-2k$ and $w=-3i+2j-k$ to find,

  $-v+2w$

Answer to Problem 78RE

Thevector is, $-9i+3j-4k$ .

Explanation of Solution

Given information:

The vectors:

  $v=3i+j-2k$ and $w=-3i+2j-k$

Formula used:

For the vectors by the additivity of $a+b$ in these components of ‘i’ added to ‘i’ and components of ‘j’ added to ‘j’,

Such as, $\left(a_{1}i+b_{1}j+c_{1}k\right)+\left(a_{2}i+b_{2}j+c_{2}k\right)=\left(a_{1}i+a_{2}i\right)+\left(b_{1}j+b_{2}j\right)+\left(c_{1}k+c_{2}k\right)$

Calculation:

Consider the vectors:

  $v=3i+j-2k$ and $w=-3i+2j-k$

Put the value of the vectors in the given expression $-v+2w$ .

Now $v=3i+j-2k$ and $w=-3i+2j-k$ .

With the help of additivity of vectors:

  $\left(a_{1}i+b_{1}j+c_{1}k\right)+\left(a_{2}i+b_{2}j+c_{2}k\right)=\left(a_{1}i+a_{2}i\right)+\left(b_{1}j+b_{2}j\right)+\left(c_{1}k+c_{2}k\right)$

Now first we calculate the value of $-v$ .

  $\begin{array}{c}-v=-\left(3i+j-2k\right)\\ =-3i-j-2k\end{array}$

Now calculate the value of $3w$ .

  $\begin{array}{c}2w=2\left(-3i+2j-k\right)\\ =-6i+4j-2k\end{array}$

Now apply these values in $-v+2w$ .

  $\begin{array}{c}-v+2w=\left(-3i-j-2k\right)+\left(-6i+4j-2k\right)\\ =-3i-6i-j+4j-2k-2k\\ =-9i+3j-4k\end{array}$

Thus, the value of the addition of vectors are: $-9i+3j-4k$ .