Which of the following is a unit vector parallel to the vector $$ v=\mathbf{i}+2 \mathbf{j}-5 \mathbf{k} $$ ? None of the others $$ \left(\frac{1}{30} \sqrt{30}, \frac{1}{15} \sqrt{30},-\frac{1}{6} \sqrt{30}\right) $$ $$ \left(\frac{2}{33} \sqrt{66},-\frac{5}{66} \sqrt{66},-\frac{5}{66} \sqrt{66}\right) $$ $$ \left(\frac{1}{9} \sqrt{6},-\frac{5}{18} \sqrt{6}, \frac{5}{18} \sqrt{6}\right) $$ $$ \left(\frac{4}{15} \sqrt{5}, \frac{1}{3} \sqrt{5},-\frac{2}{15} \sqrt{5}\right) $$

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UpStudy AI · Accepted Answer

We start with the vector
$$
v=\mathbf{i}+2\mathbf{j}-5\mathbf{k}=(1,\,2,\,-5).
$$
The magnitude of $$v$$ is
$$
\|v\|=\sqrt{1^2+2^2+(-5)^2}=\sqrt{1+4+25}=\sqrt{30}.
$$

A unit vector parallel to $$v$$ is given by
$$
\frac{v}{\|v\|}=\left(\frac{1}{\sqrt{30}},\,\frac{2}{\sqrt{30}},\,\frac{-5}{\sqrt{30}}\right).
$$

Notice that
$$
\frac{1}{\sqrt{30}}=\frac{\sqrt{30}}{30},\quad \frac{2}{\sqrt{30}}=\frac{2\sqrt{30}}{30}=\frac{\sqrt{30}}{15},\quad \frac{-5}{\sqrt{30}}=-\frac{5\sqrt{30}}{30}=-\frac{\sqrt{30}}{6}.
$$

Thus, the unit vector can also be written as
$$
\left(\frac{\sqrt{30}}{30},\,\frac{\sqrt{30}}{15},\,-\frac{\sqrt{30}}{6}\right).
$$

Comparing with the provided options, the unit vector parallel to $$v$$ is
$$
\left(\frac{1}{30}\sqrt{30},\,\frac{1}{15}\sqrt{30},\,-\frac{1}{6}\sqrt{30}\right).
$$
The unit vector parallel to $$ v = \mathbf{i} + 2\mathbf{j} - 5\mathbf{k} $$ is $$ \left(\frac{1}{30}\sqrt{30}, \frac{1}{15}\sqrt{30}, -\frac{1}{6}\sqrt{30}\right) $$.

Which of the following is a unit vector parallel to the vector ?
None of the others

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Which of the following is a unit vector parallel to the vector ? None of the others