Chapter 6, Problem 56CR

To determine

To explain:

The vector as a linear combination of the standard unit vectors $i$ and $j$ .

Answer to Problem 56CR

The vector as a linear combination of the standard unit vectors $i$ and $j$ is $7i-16j$ .

Explanation of Solution

Given information:

Initial point: $\left(-2,7\right);$ terminal point $:\left(5,-9\right)$

The initial point and terminal point of a vector are given by $\left(-2,7\right)$ and $\left(5,-9\right)$ respectively.

The objective is to write the vector as a liner combination of unit vectors $i$ and $j$ .

Let $u$ be the vector with initial point $\left(-2,7\right)$ and terminal point $\left(5,-9\right)$ .

If $P\left(p_1,p_2\right)$ and $Q\left(q_1,q_2\right)$ are the initial and terminal points of $u$ .

Then component form of $u$ is

  $\langle q_1-p_1,q_2-p_2\rangle$

Here $P=\left(-2,7\right);$ so that $p_1=-2,$ and $p_2=7$

  $Q=\left(5,-9\right),$ so that $q_1=5$ and $q_2=-9$

Component form of,

  $\begin{array}{l}u=\langle q_1-p_1,q_2-p_2\rangle \\ =\langle 5-\left(-2\right),-9-7\rangle \\ =\langle 7,-16\rangle \\ =7i-16j\end{array}$

Thus, the required vector is $7i-16j$

The result is shown graphically,

  

The above graph shows the initial point: $\left(-2,7\right)$ and terminal point $:\left(5,-9\right)$