To calculate the projection of u onto v (denoted as $\text{proj}_{\mathbf{v}} \mathbf{u}$), 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: Calculate the dot product $\mathbf{u} \cdot \mathbf{v}$

The vectors are:

$\mathbf{u} = \begin{bmatrix} 2 \\ -2 \\ 5 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} -1 \\ -4 \\ 4 \end{bmatrix}$

Now, calculate the dot product $\mathbf{u} \cdot \mathbf{v}$:

$\mathbf{u} \cdot \mathbf{v} = (2)(-1) + (-2)(-4) + (5)(4) = -2 + 8 + 20 = 26$

Step 2: Calculate $\mathbf{v} \cdot \mathbf{v}$

The dot product $\mathbf{v} \cdot \mathbf{v}$ is:

$\mathbf{v} \cdot \mathbf{v} = (-1)^2 + (-4)^2 + (4)^2 = 1 + 16 + 16 = 33$

Step 3: Calculate the projection

Now, use the formula:

$\text{proj}_{\mathbf{v}} \mathbf{u} = \frac{26}{33} \mathbf{v}$

Substitute $\mathbf{v} = \begin{bmatrix} -1 \\ -4 \\ 4 \end{bmatrix}$:

$\text{proj}_{\mathbf{v}} \mathbf{u} = \frac{26}{33} \begin{bmatrix} -1 \\ -4 \\ 4 \end{bmatrix}$

This gives:

$\text{proj}_{\mathbf{v}} \mathbf{u} = \begin{bmatrix} \frac{26}{33}(-1) \\ \frac{26}{33}(-4) \\ \frac{26}{33}(4) \end{bmatrix} = \begin{bmatrix} -\frac{26}{33} \\ -\frac{104}{33} \\ \frac{104}{33} \end{bmatrix}$

Thus, the projection is:

$\text{proj}_{\mathbf{v}} \mathbf{u} = \begin{bmatrix} -\frac{26}{33} \\ -\frac{104}{33} \\ \frac{104}{33} \end{bmatrix}$

Would you like further details or have any questions?

Here are 5 related questions:

1. How do we calculate the magnitude of the projection vector?
2. What is the geometric interpretation of the vector projection?
3. How do dot products relate to vector projections?
4. What is the angle between the vectors u and v?
5. How do we calculate the orthogonal projection?

Tip: Always verify the dot product calculations when working with projections, as small errors can affect the result significantly!