Find (𝑎,𝑏,𝑐,𝑑)
in ℝ4
such that the given set is orthogonal.
{  (−1,−1,2,−1), (−1,2,−2,−5), (5,−8,−3,−3), (𝑎,𝑏,𝑐,𝑑)  }

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:
\[
\begin{aligned}
1. & \quad a + b - 2c + d = 0 \\
2. & \quad -a + 2b - 2c - 5d = 0 \\
3. & \quad 5a - 8b - 3c - 3d = 0
\end{aligned}
\]

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?

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**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.

Learn how to find the vector (a, b, c, d) in ℝ^4 that is orthogonal to a given set of vectors (-1, -1, 2, -1), (-1, 2, -2, -5), (5, -8, -3, -3). This problem involves solving a system of linear equations using the dot product formula. Suitable for advanced undergraduate students studying linear algebra and vector spaces.

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