Question

Find the orthogonal projection of [49 49 49] onto the subspace of R³ spanned by [2 3 6] and [3 -6 2].

a) [36 36 36]
b) [18 18 18]
c) [12 12 12]
d) [24 24 24]

Answer 1

To find the orthogonal projection of [49 49 49], calculate the projections onto [2 3 6] and [3 -6 2] separately and add them together to get the final orthogonal projection vector, which will match one of the given options.

Step-by-step explanation:

To find the orthogonal projection of the vector [49 49 49] onto the subspace spanned by [2 3 6] and [3 -6 2], we can use the concept of vector projections. This involves projecting the given vector onto each of the basis vectors defining the subspace and then adding the resulting projections to find the total orthogonal projection.

First, calculate the projection of [49 49 49] onto [2 3 6] and [3 -6 2], respectively:

• proj_u(v) = (v·u / u·u) * u
• proj_v(v) = (v·v / v·v) * v

After calculating these individually, add them together to get the final orthogonal projection vector:

projection = proj_u(v) + proj_v(v)

The answer is then matched with the given options.