Linear Algebra Let u1u2u3 form an orthonormal basis for the

Linear Algebra

Let {u₁,u₂,u₃} form an orthonormal basis for the inner product space V. If x = c₁u₁ + c₂u₂ + c₃u₃ is a vector with the properties that ||x|| = 5, <u₁,x> = 4 and x is perpendicular to u₂, then determine the possible values of c₁,c₂, and c₃. Please justify your answers.

Solution

given ||X|| = 5

X = c1u1 + c2u2 +c3u3

||X||² = c1² + c2² + c3³   = 25

<u1,X> = 4

dot product of u1 and X is 4

<u1, c1u1 + c2u2 +c3u3 > = c1 = 4

X is perpendicular to u2 ===>> <u2 , X> = 0

<u2, X> = <u2, c1u1 + c2u2 +c3u3> = c2

so c2 = 0

c1² + c2² + c3³   = 25

4² + 0 + c3² = 25

c3² = 25 - 16 = 9

So possible values of c3 is 3 or -3

So ( c1 , c2 , c3 ) = (4, 0 , 3) or (4, 0 ,-3)