Chapter 11, Problem 1RE

(a)

To determine

To Find:Component form of $\text{3u-4v}$ .

Answer to Problem 1RE

The required component of the vector is $\left(1,14\right)$ .

Explanation of Solution

Given Information:

  $\begin{array}{l}u=\left(-3,4\right)\\ v=\left(2,-5\right)\end{array}$

Calculation:

Component form of the vector is

  $\begin{array}{l}\text{3u-4v}\\ \text{=3}\times \left(3,-2\right)-4\times \left(2,-5\right)\\ =\left(9,-6\right)-\left(8,-20\right)\\ =\left(9-8,-6+20\right)\\ =\left(1,14\right)\end{array}$

Conclusion:

Hence, the required component of the vector is $\left(1,14\right)$ .

(b)

To determine

To Find: The magnitude of $\text{3u-4v}$ .

Answer to Problem 1RE

The required magnitude of the vector is $\left|v\right|=\sqrt{197}$ .

Explanation of Solution

Given Information:

  $\begin{array}{l}u=\left(-3,4\right)\\ v=\left(2,-5\right)\end{array}$

Component of $\text{3u-4v}$ is $\left(1,14\right)$

Formula Used:

Magnitude formula

  $\left|v\right|=\sqrt{a^2+b^2}$  ...... (1)

Here, $\left|v\right|$ is the magnitude, $a$ and $b$ are two points.

Calculation:

Substitute $\left(1,14\right)$ for $a,b$ in equation (1)

  $\begin{array}{c}\left|v\right|=\sqrt{{\left(1\right)}^2+{\left(14\right)}^2}\\ =\sqrt{197}\end{array}$

Conclusion:

Hence, the required magnitude of the vector is $\left|v\right|=\sqrt{197}$ .