Question

Question: What is the magnitude of the vector $\mathit{v}$, where $\mathit{v}\mathbf{=}\left(8\times {10}^6,0,-2\times {10}^7\right)\mathbf{}\mathit{m}\mathbf{/}\mathit{s}\mathbf{}\mathbf{}\mathbf{?}$

Short answer

Answer

Magnitude of given vector is $\mathbf{21}\mathbf{.}\mathbf{54}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$

Step-by-step solution

Formula used

Magnitude of a vector $\mathit{a}$having coordinates $\left(x,y,z\right)$is given by$\left|a\right|\mathbf{=}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}}$. The magnitude of a vector gives the length of the vector so, basically magnitude is used to find the length of a vector between the initial and the final point of the vector.

Calculation of magnitude

Substitute $\mathit{x}\mathbf{=}\mathbf{8}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{}\mathbf{}\mathbf{,}\mathbf{}\mathbf{}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{}\mathbf{}$and $\mathit{z}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{7}}$into the formula of magnitude to get the magnitude of the given vector.

$$\begin{gathered} \left|v\right|\mathbf{=}\sqrt{{\left(8\times {10}^6\right)}^{\mathbf{2}}\mathbf{+}}{\mathbf{0}}^{\mathbf{2}}\mathbf{+}{\left(-2\times {10}^7\right)}^{\mathbf{2}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\sqrt{\left(64\times {10}^{12}\right)\mathbf{+}\mathbf{0}\mathbf{+}\left(4\times {10}^{14}\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\sqrt{\mathbf{464}\mathbf{\times }{\mathbf{10}}^{\mathbf{12}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{21}\mathbf{.}\mathbf{54}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{}\mathit{m}\mathbf{/}\mathit{s} \end{gathered}$$

Thus, the magnitude of the given vector $\left|v\right|$is $\mathbf{21}\mathbf{.}\mathbf{54}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}\mathbf{}}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$.