Answer:
- X(4, 8, 8)
- Y'(-2, -1, 2)
- Z'(-2, 2, -1)
Explanation:
You want to perform the Gram-Schmidt process on vectors X(4, 8, 8), Y(-4, -5, -2), and Z(-6, -3, -3) to produce a set of orthogonal vectors.
Orthogonal vectors
A second vector with a direction distinct from a first vector can be adjusted so that it is orthogonal to the first by removing the component of it that is in the direction of the first vector. The amount of this projection of the second vector y on first vector x can be computed as ...
Proj(x, y) = ((x•y)/(x•x))x
where x•y is the dot product of vectors x and y.
Gram-Schmidt process
The Gram-Schmidt process systematically removes the projection of each vector on the ones before, resulting in a set of orthogonal vectors. The first attachment gives the idea of it.
The second attachment shows the computation of the orthogonal vector set. The end result in that attachment is the set of vectors (lists in this calculator) L1, L4, L5.
- L1 = (4, 8, 8) . . . . . same as X
- L4 = (-2, -1, 2) . . . . derived from Y, orthogonal to L1
- L5 = (-2, 2, -1) . . . . derived from Z, orthogonal to L1 and L4
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Additional comment
The third attachment shows the calculator definition of the Proj(x, y) function we used. It is the same as that shown above.
The last line in the second attachment shows the resulting vectors are orthogonal (mutual dot products are zero).
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