Question

Project b = (0, 3, 0) onto each of the orthonormal vectors a₁ = (2/3, 2/3, −1/3) and

a₂ = (−1/3, 2/3, 2/3) and then find its projection p onto the plane of a₁ and a₂.

Answer 1

To find the projection of vector b onto a₁ and a₂, we can use the formula proja(b) = (b · a) / (a · a) * a. The projections onto a₁ and a₂ are (4/3, 4/3, -2/3) and (-2/3, 4/3, 4/3) respectively. The projection of b onto the plane of a₁ and a₂ is (10/3, 10/3, -2/3)

Step-by-step explanation:

To project vector b onto each of the orthonormal vectors a₁ and a₂, we can use the formula for projection: proja(b) = (b · a) / (a · a) * a. First, we will calculate the projections onto a₁ and a₂:

proja₁(b) = (b · a₁) / (a₁ · a₁) * a₁ = ((0)(2/3) + (3)(2/3) + (0)(-1/3)) / ((2/3)(2/3) + (2/3)(2/3) + (-1/3)(-1/3)) * (2/3, 2/3, -1/3) = (4/3, 4/3, -2/3)

proja₂(b) = (b · a₂) / (a₂ · a₂) * a₂ = ((0)(-1/3) + (3)(2/3) + (0)(2/3)) / ((-1/3)(-1/3) + (2/3)(2/3) + (2/3)(2/3)) * (-1/3, 2/3, 2/3) = (-2/3, 4/3, 4/3)

To find the projection p onto the plane of a₁ and a₂, we can subtract the individual projections from vector b:

p = b - proja₁(b) - proja₂(b) = (0, 3, 0) - (4/3, 4/3, -2/3) - (-2/3, 4/3, 4/3) = (10/3, 10/3, -2/3)