Let p1 = − 22 22 and p2 = 22 22. If the matrix P is orthogonal, show that the column vectors of the matrix form an orthonormal set. (If the matrix is not orthogonal, enter NOT ORTHOGONAL.)Find p1 · p2
Question
Let p1 = − 22 22 and p2 = 22 22. If the matrix P is orthogonal, show that the column vectors of the matrix form an orthonormal set. (If the matrix is not orthogonal, enter NOT ORTHOGONAL.)Find p1 · p2
Solution
To determine if the column vectors of the matrix P form an orthonormal set, we need to check two conditions: orthogonality and normalization.
- Orthogonality: To check if the column vectors of P are orthogonal, we need to calculate the dot product of each pair of column vectors and verify if the result is zero.
Let's calculate the dot product of p1 and p2: p1 · p2 = (-22 * 22) + (22 * 22) = -484 + 484 = 0
Since the dot product of p1 and p2 is zero, the column vectors of P are orthogonal.
- Normalization: To check if the column vectors of P are normalized, we need to calculate the magnitude (or length) of each column vector and verify if the result is 1.
Let's calculate the magnitude of p1: ||p1|| = sqrt((-22)^2 + 22^2) = sqrt(484 + 484) = sqrt(968) ≈ 31.1127
Let's calculate the magnitude of p2: ||p2|| = sqrt(22^2 + 22^2) = sqrt(484 + 484) = sqrt(968) ≈ 31.1127
Since both magnitudes are approximately equal to 31.1127, the column vectors of P are normalized.
Therefore, since the column vectors of the matrix P satisfy both the orthogonality and normalization conditions, we can conclude that they form an orthonormal set.
Additionally, the dot product of p1 and p2 is 0.
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