Find a vector $$ (\vec{u}) $$ with magnitude 2 in the direction of $$ \vec{v}=\langle 1,4\rangle $$ Give EXACT answer. You do NOT have to simplify your radicals! $$ \vec{u}= $$ Submit Question

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UpStudy AI · Accepted Answer

We are given the vector $$ \vec{v} = \langle 1,4 \rangle $$ and we need a vector $$ \vec{u} $$ of magnitude 2 in the direction of $$ \vec{v} $$.

**Step 1.** Calculate the magnitude (length) of $$ \vec{v} $$:

$$
\|\vec{v}\| = \sqrt{1^2+4^2}=\sqrt{1+16}=\sqrt{17}.
$$

**Step 2.** Find the unit vector in the direction of $$ \vec{v} $$:

$$
\hat{v} = \frac{\vec{v}}{\|\vec{v}\|} = \frac{1}{\sqrt{17}} \langle 1,4 \rangle = \left\langle \frac{1}{\sqrt{17}}, \frac{4}{\sqrt{17}} \right\rangle.
$$

**Step 3.** Multiply the unit vector by 2 to obtain $$ \vec{u} $$ with the desired magnitude:

$$
\vec{u} = 2 \hat{v} = 2 \left\langle \frac{1}{\sqrt{17}}, \frac{4}{\sqrt{17}} \right\rangle = \left\langle \frac{2}{\sqrt{17}}, \frac{8}{\sqrt{17}} \right\rangle.
$$

Thus, the vector $$ \vec{u} $$ is

$$
\vec{u}= \left\langle \frac{2}{\sqrt{17}}, \frac{8}{\sqrt{17}} \right\rangle.
$$
$$ \vec{u} = \left\langle \frac{2}{\sqrt{17}}, \frac{8}{\sqrt{17}} \right\rangle $$

Find a vector with magnitude 2 in the direction of
Give EXACT answer. You do NOT have to simplify your radicals!
Submit Question

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Find a vector with magnitude 2 in the direction of Give EXACT answer. You do NOT have to simplify your radicals! Submit Question