Chapter 5.CM, Problem 1CM

To determine

(a)

To find:

The value of $v+w$ and sketch the vector.

Answer to Problem 1CM

Solution:

The value of $v+w$ is $\left(3,-7\right)$ and the graph of the vector is shown in figure $\left(1\right)$.

Explanation of Solution

Given:

The vectors are as follows.

$\begin{array}{rll}v& =& \left(1,-2\right)\\ w& =& \left(2,-5\right)\end{array}$

Approach:

Add individual elements of the two vectors.

Calculation:

Find the value of $v+w$.

$\begin{array}{rll}v+w& =& \left(1,-2\right)+\left(2,-5\right)\\ & =& \left(1+2,-2-5\right)\\ & =& \left(3,-7\right)\end{array}$

The following figure shows the graph of each vector.

Figure $\left(1\right)$

Conclusion:

Hence, the value of $v+w$ is $\left(3,-7\right)$ and the graph of the vector is shown in figure $\left(1\right)$.

To determine

(b)

To find:

The value of $3v$.

Answer to Problem 1CM

Solution:

The value of $3v$ is $\left(3,-6\right)$ and the graph of the vector is shown in figure $\left(2\right)$.

Explanation of Solution

Calculation:

Find $3v$.

$\begin{array}{rll}3v& =& 3\left(1,-2\right)\\ & =& \left(3,-6\right)\end{array}$

The following figure shows the graph of each vector.

Figure $\left(2\right)$

Conclusion:

Hence, the value of $3v$ is $\left(3,-6\right)$ and the graph of the vector is shown in figure $\left(2\right)$.

To determine

(c)

To find:

The value of $2v-4w$.

Answer to Problem 1CM

Solution:

The value of $2v-4w$ is $\left(-6,16\right)$ and graph of the vector is shown in figure $\left(3\right)$.

Explanation of Solution

Calculation:

Find $2v-4w$.

$\begin{array}{rll}2v-4w& =& 2\left(1,-2\right)-4\left(2,-5\right)\\ & =& \left(2,-4\right)-\left(8,-20\right)\\ & =& \left(2-8,-4+20\right)\\ & =& \left(-6,16\right)\end{array}$

Therefore, the value of $2v-4w$ is $\left(-6,16\right)$.

The following figure shows the graph of each vector.

Figure $\left(3\right)$

Conclusion:

Hence, the value of $2v-4w$ is $\left(-6,16\right)$ and graph of the vector is shown in figure $\left(3\right)$.