From the image, the question deals with vector operations. Let's break down the problem:

Question Details:

1. Given vectors:

   • $\mathbf{u} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$,
   • $\mathbf{v} = \begin{pmatrix} 1 \\ 5 \end{pmatrix}$.

2. Tasks: a) Determine $\mathbf{t} = \mathbf{u} + \mathbf{v}$.
   b) Find a vector $\mathbf{p}$ such that it is orthogonal to both $\mathbf{u}$ and $\mathbf{t}$.

Solution:

(a) Determine $\mathbf{t}$:

$\mathbf{t} = \mathbf{u} + \mathbf{v} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 1 \\ 5 \end{pmatrix} = \begin{pmatrix} 2 + 1 \\ 3 + 5 \end{pmatrix} = \begin{pmatrix} 3 \\ 8 \end{pmatrix}.$

So, $\mathbf{t} = \begin{pmatrix} 3 \\ 8 \end{pmatrix}$.

(b) Find vector $\mathbf{p}$ orthogonal to $\mathbf{u}$ and $\mathbf{t}$:

A vector $\mathbf{p} = \begin{pmatrix} x \\ y \end{pmatrix}$ is orthogonal to $\mathbf{u}$ and $\mathbf{t}$ if: $\mathbf{u} \cdot \mathbf{p} = 0 \quad \text{and} \quad \mathbf{t} \cdot \mathbf{p} = 0.$ Here, the dot product is defined as: $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2.$

1. Orthogonal to $\mathbf{u}$: \mathbf{u} \cdot \mathbf{p} = 2x + 3y = 0. \tag{1}

2. Orthogonal to $\mathbf{t}$: \mathbf{t} \cdot \mathbf{p} = 3x + 8y = 0. \tag{2}

We now solve this system of linear equations:

• From (1): $2x + 3y = 0 \implies x = -\frac{3}{2}y.$
• Substituting $x = -\frac{3}{2}y$ into (2): $3\left(-\frac{3}{2}y\right) + 8y = 0 \implies -\frac{9}{2}y + 8y = 0 \implies \frac{7}{2}y = 0.$ Thus, $y = 0$, and substituting back, $x = 0$.

This suggests that there is an issue with orthogonality across $\mathbf{u}$ and $\mathbf{t}$, since their span covers the 2D space. Hence, no non-zero vector orthogonal to both exists in 2D.

Would you like further assistance or clarifications on this? Here are 5 follow-up questions:

1. How does orthogonality relate to the span of vectors in 2D or higher dimensions?
2. Could you explain why a solution might not exist for orthogonality constraints in specific cases?
3. How can cross products help to find orthogonal vectors in 3D?
4. What does the geometric interpretation of $\mathbf{u} \cdot \mathbf{v} = 0$ imply?
5. How do you verify solutions of systems of equations like these?

Tip: Always confirm the independence of given vectors before solving