Question

Find an orthogonal matrix P whose first row is (2/3,−1/3,2/3).

Answer 1

To find an orthogonal matrix P with a given first row, we need to find two other rows that are orthogonal to the first row. We can set up equations using the dot product and solve for the other rows.

Step-by-step explanation:

To find an orthogonal matrix P whose first row is (2/3, -1/3, 2/3), we need to find two other rows that are orthogonal to the given row. Let's assign variables a and b to the second and third rows respectively. Since the dot product of two orthogonal vectors is zero, we can set up two equations:

`(2/3)(a) + (-1/3)(b) = 0`

`(2/3)(2/3) + (-1/3)(2/3) + (2/3)(b) = 0`

Solving these equations, we get a = (-1/2)(2/3, -1/3, 2/3) and b = (4/3, -2/3, -2/3). Therefore, the orthogonal matrix P is:

P = [(2/3, -1/3, 2/3), (-1/2)(2/3, -1/3, 2/3), (4/3, -2/3, -2/3)]