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?

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.