# Answer to Question #5213 in Linear Algebra for Jill

Question #5213

Find the orthogonal projection of u on a

u = (6, 6, -5) and a = (1, -2, 0)

Give the exact answer, using fractions if necessary.

projau = (?, ?, ?)

Thank you for your help!

u = (6, 6, -5) and a = (1, -2, 0)

Give the exact answer, using fractions if necessary.

projau = (?, ?, ?)

Thank you for your help!

Expert's answer

u = (6, 6, -5) and a = (1, -2, 0).

proj = u*cosA,

where A is the angle between vectors u and a.

u*a = |u|*|a|*cosA,

cosA = u*a/(|u|*|a|),

u*a = 6*1 - 6*2 - 5*0 = -6,

|u| = sqrt(6^2 + 6^2 + 5^2) = sqrt(97),

|a| = sqrt(5).

proj = (6, 6, -5)*(-6/sqrt(97*5)) = (-1.63, -1.63, 1.36).

proj = u*cosA,

where A is the angle between vectors u and a.

u*a = |u|*|a|*cosA,

cosA = u*a/(|u|*|a|),

u*a = 6*1 - 6*2 - 5*0 = -6,

|u| = sqrt(6^2 + 6^2 + 5^2) = sqrt(97),

|a| = sqrt(5).

proj = (6, 6, -5)*(-6/sqrt(97*5)) = (-1.63, -1.63, 1.36).

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