Questions: Find the unit vector u with the same direction as t = -7 i + 10 j + k. Simplify any radicals. u = □ i + □ j + □ k

Find the unit vector u with the same direction as t = -7 i + 10 j + k. Simplify any radicals.

u = □ i + □ j + □ k
Transcript text: Find the unit vector $\mathbf{u}$ with the same direction as $\mathbf{t}=-7 \mathbf{i}+10 \mathbf{j}+\mathbf{k}$. Simplify any radicals. \[ \mathbf{u}=\square \mathbf{i}+\square \mathbf{j}+\square \mathbf{k} \]
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Solution

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Solution Steps

Step 1: Find the Magnitude of t\mathbf{t}

Calculate the magnitude of the vector t=7i+10j+k\mathbf{t} = -7 \mathbf{i} + 10 \mathbf{j} + \mathbf{k} using the formula for the magnitude of a vector:

t=(7)2+102+12 \|\mathbf{t}\| = \sqrt{(-7)^2 + 10^2 + 1^2}

t=49+100+1 \|\mathbf{t}\| = \sqrt{49 + 100 + 1}

t=150 \|\mathbf{t}\| = \sqrt{150}

t=25×6=56 \|\mathbf{t}\| = \sqrt{25 \times 6} = 5\sqrt{6}

Step 2: Divide Each Component by the Magnitude

To find the unit vector u\mathbf{u}, divide each component of t\mathbf{t} by its magnitude t\|\mathbf{t}\|:

u=(756)i+(1056)j+(156)k \mathbf{u} = \left(\frac{-7}{5\sqrt{6}}\right) \mathbf{i} + \left(\frac{10}{5\sqrt{6}}\right) \mathbf{j} + \left(\frac{1}{5\sqrt{6}}\right) \mathbf{k}

Step 3: Simplify Each Component

Simplify each component of the unit vector:

u=(756)i+(1056)j+(156)k \mathbf{u} = \left(\frac{-7}{5\sqrt{6}}\right) \mathbf{i} + \left(\frac{10}{5\sqrt{6}}\right) \mathbf{j} + \left(\frac{1}{5\sqrt{6}}\right) \mathbf{k}

Rationalize the denominators:

u=(7630)i+(10630)j+(630)k \mathbf{u} = \left(\frac{-7\sqrt{6}}{30}\right) \mathbf{i} + \left(\frac{10\sqrt{6}}{30}\right) \mathbf{j} + \left(\frac{\sqrt{6}}{30}\right) \mathbf{k}

Simplify the fractions:

u=(7630)i+(63)j+(630)k \mathbf{u} = \left(\frac{-7\sqrt{6}}{30}\right) \mathbf{i} + \left(\frac{\sqrt{6}}{3}\right) \mathbf{j} + \left(\frac{\sqrt{6}}{30}\right) \mathbf{k}

Final Answer

u=7630i+63j+630k\mathbf{u} = \frac{-7\sqrt{6}}{30} \mathbf{i} + \frac{\sqrt{6}}{3} \mathbf{j} + \frac{\sqrt{6}}{30} \mathbf{k}

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