To find $(a, b, c, d)$ such that the given set is orthogonal, we need to ensure that the vector $(a, b, c, d)$ is orthogonal to each of the other three vectors in the set.

Given vectors: $\mathbf{v_1} = (-1, -1, 2, -1), \quad \mathbf{v_2} = (-1, 2, -2, -5), \quad \mathbf{v_3} = (5, -8, -3, -3)$

The vector $(a, b, c, d)$ must satisfy: $\mathbf{v_4} \cdot \mathbf{v_1} = 0$ $\mathbf{v_4} \cdot \mathbf{v_2} = 0$ $\mathbf{v_4} \cdot \mathbf{v_3} = 0$

Where $\mathbf{v_4} = (a, b, c, d)$.

We calculate each dot product:

1. For $\mathbf{v_4} \cdot \mathbf{v_1} = 0$: $(-1)a + (-1)b + 2c + (-1)d = 0 \quad \Rightarrow \quad -a - b + 2c - d = 0 \quad \Rightarrow \quad a + b - 2c + d = 0$

2. For $\mathbf{v_4} \cdot \mathbf{v_2} = 0$: $(-1)a + 2b + (-2)c + (-5)d = 0 \quad \Rightarrow \quad -a + 2b - 2c - 5d = 0$

3. For $\mathbf{v_4} \cdot \mathbf{v_3} = 0$: $5a + (-8)b + (-3)c + (-3)d = 0 \quad \Rightarrow \quad 5a - 8b - 3c - 3d = 0$

So, we have the following system of equations:

1. & \quad a + b - 2c + d = 0 \\ 2. & \quad -a + 2b - 2c - 5d = 0 \\ 3. & \quad 5a - 8b - 3c - 3d = 0 \end{aligned}$$ Next, we solve this system of linear equations to find $$a$$, $$b$$, $$c$$, and $$d$$. I'll solve these equations now.The solution to the system of equations is: $$a = \frac{-123d}{31}, \quad b = \frac{-60d}{31}, \quad c = \frac{-76d}{31}$$ Here, $$d$$ can be any real number. Therefore, the vector $$(a, b, c, d)$$ can be expressed as: $$\left(\frac{-123d}{31}, \frac{-60d}{31}, \frac{-76d}{31}, d\right)$$ This vector is orthogonal to the other three vectors in the set for any value of $$d \in \mathbb{R}$$. Would you like more details or have any questions? --- **Relative Questions:** 1. How do you verify that the vectors are orthogonal? 2. What happens if $$d = 0$$ in this solution? 3. Can you find the magnitude of the vector $$(a, b, c, d)$$ for a specific $$d$$? 4. What is the importance of orthogonal vectors in vector spaces? 5. How would the solution change if one of the original vectors was altered? **Tip:** Orthogonality in vector spaces is crucial for simplifying problems in various fields such as physics, engineering, and computer science, particularly in areas involving projections and decompositions.