To solve the problem:

1. Vector projection formula: The projection of $\mathbf{u}$ onto $\mathbf{v}$, denoted as $\mathbf{u}_{\mathbf{v}}$, is calculated as:

   $\mathbf{u}_{\mathbf{v}} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v}$

   Here:

   • $\mathbf{u} = (1, 1, 2)$
   • $\mathbf{v} = (1, 0, -1)$
   • $\mathbf{u} \cdot \mathbf{v}$ is the dot product of $\mathbf{u}$ and $\mathbf{v}$,
   • $\|\mathbf{v}\|^2$ is the squared magnitude of $\mathbf{v}$.

2. Orthogonal component: After finding $\mathbf{u}_{\mathbf{v}}$, the orthogonal component $\mathbf{u} - \mathbf{u}_{\mathbf{v}}$ will be calculated. This vector should be perpendicular to $\mathbf{v}$, meaning their dot product will be 0.

Step-by-step Solution:

Step 1: Compute $\mathbf{u} \cdot \mathbf{v}$

$\mathbf{u} \cdot \mathbf{v} = (1)(1) + (1)(0) + (2)(-1) = 1 + 0 - 2 = -1$

Step 2: Compute $\|\mathbf{v}\|^2$

$\|\mathbf{v}\|^2 = (1)^2 + (0)^2 + (-1)^2 = 1 + 0 + 1 = 2$

Step 3: Calculate $\mathbf{u}_{\mathbf{v}}$

$\mathbf{u}_{\mathbf{v}} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v} = \frac{-1}{2}(1, 0, -1) = \left(-\frac{1}{2}, 0, \frac{1}{2}\right)$

Step 4: Compute $\mathbf{u} - \mathbf{u}_{\mathbf{v}}$

$\mathbf{u} - \mathbf{u}_{\mathbf{v}} = (1, 1, 2) - \left(-\frac{1}{2}, 0, \frac{1}{2}\right) = \left(1 + \frac{1}{2}, 1 - 0, 2 - \frac{1}{2}\right) = \left(\frac{3}{2}, 1, \frac{3}{2}\right)$

Step 5: Verify orthogonality

To confirm that $\mathbf{u} - \mathbf{u}_{\mathbf{v}}$ is perpendicular to $\mathbf{v}$, compute their dot product: $\left(\frac{3}{2}, 1, \frac{3}{2}\right) \cdot (1, 0, -1) = \left(\frac{3}{2}\right)(1) + (1)(0) + \left(\frac{3}{2}\right)(-1) = \frac{3}{2} - \frac{3}{2} = 0$

Since the dot product is 0, $\mathbf{u} - \mathbf{u}_{\mathbf{v}}$ is orthogonal to $\mathbf{v}$.

Final Results:

1. $\mathbf{u}_{\mathbf{v}} = \left(-\frac{1}{2}, 0, \frac{1}{2}\right)$
2. $\mathbf{u} - \mathbf{u}_{\mathbf{v}} = \left(\frac{3}{2}, 1, \frac{3}{2}\right)$, which is perpendicular to $\mathbf{v}$.

Would you like further details on any step?

Relative Questions:

1. What is the significance of vector projection in physics or engineering?
2. How do you calculate the magnitude of a vector given its components?
3. What are some real-world examples of orthogonal vectors?
4. How is the concept of orthogonality used in computer graphics?
5. Can projections be extended to spaces with more than three dimensions?

Tip: Always double-check dot products when working with projections; a small mistake can change the entire outcome.