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 the values of \((a, b, c, d)\) such that the given set of vectors is orthogonal, we need to ensure that the dot product of \((a, b, c, d)\) with each of the three other vectors is zero. The given vectors are:

\[
v_1 = (-1, -1, 2, -1), \quad v_2 = (-1, 2, -2, -5), \quad v_3 = (5, -8, -3, -3), \quad v_4 = (a, b, c, d)
\]

The dot products are calculated as follows:

1. **Dot product with \(v_1\)**:
\[
v_1 \cdot v_4 = -1a - 1b + 2c - 1d = 0
\]
\[
-a - b + 2c - d = 0 \quad \text{(Equation 1)}
\]

2. **Dot product with \(v_2\)**:
\[
v_2 \cdot v_4 = -1a + 2b - 2c - 5d = 0
\]
\[
-a + 2b - 2c - 5d = 0 \quad \text{(Equation 2)}
\]

3

Explore how to find the values of (a, b, c, d) in ℝ4 such that a given set of vectors forms an orthogonal set. This problem involves calculating dot products and ensuring orthogonality in vector spaces.

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