To compute the projection of $\mathbf{u}$ onto $\mathbf{v}$, we use the formula:

$\text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}$

Step 1: Compute the dot products $\mathbf{u} \cdot \mathbf{v}$ and $\mathbf{v} \cdot \mathbf{v}$

$\mathbf{u} \cdot \mathbf{v} = (-1)(2) + (2)(-2) + (1)(1) = -2 - 4 + 1 = -5$

$\mathbf{v} \cdot \mathbf{v} = (2)^2 + (-2)^2 + (1)^2 = 4 + 4 + 1 = 9$

Step 2: Compute the projection

$\text{proj}_{\mathbf{v}} \mathbf{u} = \frac{-5}{9} \mathbf{v} = \frac{-5}{9} \langle 2, -2, 1 \rangle$

Step 3: Simplify

$\text{proj}_{\mathbf{v}} \mathbf{u} = \left\langle \frac{-10}{9}, \frac{10}{9}, \frac{-5}{9} \right\rangle$

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Would you like a step-by-step visualization of the vector projection?

Here are 5 related questions to deepen your understanding:

1. How do dot products relate to vector projection?
2. What is the geometric meaning of the projection of one vector onto another?
3. How would you compute the projection if vectors were expressed symbolically instead of numerically?
4. How does the length of the projection vector compare to the original vector?
5. What happens when the two vectors are orthogonal?

Tip: The projection vector always points in the direction of the base vector $\mathbf{v}$.