Questions: 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. \[ \mathbf{u}=\square \mathbf{i}+\square \mathbf{j}+\square \mathbf{k} \]

Solution

Solution Steps

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

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

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

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

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

\[ \|\mathbf{t}\| = \sqrt{25 \times 6} = 5\sqrt{6} \]

Step 2: Divide Each Component by the Magnitude

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

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

Step 3: Simplify Each Component

Simplify each component of the unit vector:

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

Rationalize the denominators:

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

Simplify the fractions:

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