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To find the unit vector in the same direction as **v = 2i + 2j - k**, we need to normalize this vector. The unit vector \(\hat{v}\) is given by the formula:

\[
\hat{v} = \frac{\vec{v}}{|\vec{v}|}
\]

Where \(|\vec{v}|\) is the magnitude of vector \(\vec{v}\), and it's calculated as:

\[
|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}
\]

For \(\vec{v} = 2i + 2j - k\), we have:
- \(v_x = 2\)
- \(v_y = 2\)
- \(v_z = -1\)

Now, calculate the magnitude:

\[
|\vec{v}| = \sqrt{2^2 + 2^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3
\]

Now, divide each component of the vector by the magnitude to get the unit vector:

\[
\hat{v} = \frac{1}{3}(2i + 2j - k) = \frac{2}{3}i + \frac{2}{3}j - \frac{1}{3}k
\]

Thus, the unit vector in the same direction as **v** is:

\[
\hat{v} = \frac{2}{3}i + \frac{2}{3}j - \frac{1}{3}k
\]

Would you like more details, or have any questions on this?

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**Related Questions:**
1. What is the formula for finding the unit vector in general?
2. How do you calculate the magnitude of a vector in 3D space?
3. Can the direction of a vector be changed by normalizing it?
4. What are the applications of unit vectors in physics and engineering?
5. How do you compute the dot product between two vectors?

**Tip:** The unit vector is often used to indicate direction without affecting magnitude, making it especially useful in physics problems involving direction or force.

Find the unit vector that has the same direction as v = 2i + 2j - k.

This tutorial explains how to find the unit vector in the direction of v = 2i + 2j - k. Follow the step-by-step solution using vector algebra and key formulas like the unit vector formula and magnitude calculation. Ideal for students in Grades 10-12.

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