Question

Find the length of each of the vectors$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}$In exercises 1 through 3.

3.$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}\mathbf{=}\mathbf{\left[}\begin{array}{c}\mathbf{2}\\ \mathbf{3}\\ \mathbf{4}\\ \mathbf{5}\end{array}\mathbf{\right]}$

Short answer

The length of the vector is about 7.35.

Step-by-step solution

Consider the definition below

The length(or magnitude or norm) of a vector$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}\mathbf{}\mathbf{in}\mathbf{}{\mathit{R}}^{\mathbf{n}}\mathbf{}\mathbf{is}\mathbf{}\mathbf{\left|}\mathbf{|}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}\mathbf{}\mathbf{|}\mathbf{\right|}\mathbf{=}\sqrt{\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}\mathbf{}\mathbf{.}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{v}}\mathbf{}}$

Use the definition

By using the above definition the length of the vector will be.

$$\begin{gathered} \left||\stackrel{\rightharpoonup }{\mathrm{v}}|\right|=\sqrt{\stackrel{\rightharpoonup }{\mathrm{v}}.\stackrel{\rightharpoonup }{\mathrm{v}}} \\ =\sqrt{\left[2345\right]\left[\begin{array}{c}2\\ 3\\ 4\\ 5\end{array}\right]} \\ =\sqrt{2-2+3-3+4-4+5-5} \\ =\sqrt{4+9+16+25} \\ =\sqrt{54} \\ \approx 7.35 \end{gathered}$$

Hence, the value of the vector is about 7.35.