Question

If $\overrightarrow{\mathbf{B}}$is added to $\overrightarrow{\mathbf{A}}$, the result is$\mathbf{6}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}$ . If $\overrightarrow{\mathbf{B}}$is subtracted fromrole="math" localid="1656310989938" $\overrightarrow{\mathbf{A}}$ , the result is $\mathbf{-}\mathbf{4}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{7}\mathbf{.}\mathbf{0}\hat{\mathbf{j}}$. What is the magnitude of role="math" localid="1656311031101" $\overrightarrow{\mathbf{A}}$?

Short answer

The magnitude of$\overrightarrow{\mathrm{A}}$ is 4.1

Step-by-step solution

To understand the concept

This problem deals with the addition and subtraction of vectors.

To find the magnitude of A→

Given,

$$\begin{gathered} \overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=6.0\hat{\mathrm{i}}+1.0\hat{\mathrm{j}}\left(\mathrm{i}\right) \\ \overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}}=-4.0\hat{\mathrm{i}}+7.0\hat{\mathrm{j}}\left(\mathrm{ii}\right) \end{gathered}$$

Adding equations (i) and (ii),

$$\begin{gathered} (\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}})+(\overrightarrow{\mathrm{A}}-\overrightarrow{\mathrm{B}})=(6.0\hat{\mathrm{i}}+1.0\hat{\mathrm{j}})+(-4.0\hat{\mathrm{i}}+7.0\mathrm{j}) \\ 2\overrightarrow{\mathrm{A}}⃗=(2.0\hat{\mathrm{i}}+8.0\hat{\mathrm{j}}) \end{gathered}$$

Thus

$\overrightarrow{\mathrm{A}}=(1.0\hat{\mathrm{i}}+4.0\hat{\mathrm{j}})$ (iii)

Using equation (iii), the magnitude of $\overrightarrow{\mathrm{A}}$is then given by

$$\begin{gathered} \mathrm{A}=|\mathrm{A}⃗|=\sqrt{{{\mathrm{A}}_{\mathrm{x}}}^2+{{\mathrm{A}}_{\mathrm{y}}}^2} \\ \mathrm{A}=\sqrt{(1.0{)}^2+(4.0{)}^2} \\ \mathrm{A}=4.1 \end{gathered}$$