Chapter 6.3, Problem 99E

To determine

The error in finding the component form of the vector $\overrightarrow{u}$ that has initial point $\left(-3,4\right)$ and terminal point $\left(6,-1\right)$

Answer to Problem 99E

The error in calculation is of the signs of vector components. The correct value of vector is $\overrightarrow{u}=\langle \text{9,-5}\rangle$ .

Explanation of Solution

Given data: $y$

The components calculated are ${\overrightarrow{u}}_1=-3-6=-9$ and ${\overrightarrow{u}}_2=4-\left(-1\right)=5$ . So $\overrightarrow{u}=\langle -9,5\rangle$ .

Method/Formula used:

Let $A\left(x_1,y_1\right)$ and $B\left(x_2,y_2\right)$ be initial and terminal points of vector $\overrightarrow{AB}$ . Then vector $\overrightarrow{OA}=x_1\text{i}+y_1\text{j}$ and $\overrightarrow{OB}=x_2\text{i}+y_2\text{j}$ . Therefore, the vector $\overrightarrow{AB}$ is given by $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=\left(x_2-x_1\right)\text{i}+\left(y_2-y_1\right)\text{j}$ or $\overrightarrow{AB}=\langle \left(x_2-x_1\right),\left(y_2-y_1\right)\rangle$

Calculation:

Let the initial and terminal points of vector $\overrightarrow{u}$ are $A\left(-3,4\right)$ and $B\left(6,-1\right)$ respectively.

Then $\overrightarrow{OA}=\text{-3i+4j}$ and $\overrightarrow{OB}=6\text{i-j}$ . Then $\overrightarrow{u}=\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$

  $\Rightarrow \overrightarrow{u}=\left(\text{6i-j}\right)\text{-}\left(\text{-3i+4j}\right)=\text{9i-5j}$

  $\Rightarrow \overrightarrow{u}=\langle \text{9,-5}\rangle$

Thus, the correct value of vector is $\overrightarrow{u}=\langle \text{9,-5}\rangle$

Therefore, the error in calculation is of the signs of vector components. The correct value of vector is $\overrightarrow{u}=\langle \text{9,-5}\rangle$ .