Question

10.12. Orthogonal component Consider the following vectors: mathbf{w}_1 = egin{bmatrix} -3 \ -4 \ 0 end{bmatrix}, quad mathbf{w}_2 = egin{bmatrix} 52 \ -39 \ -75 end{bmatrix}, quad mathbf{v} = egin{bmatrix} 0 \ -1 \ 1 end{bmatrix} The set mathcal{B} = {mathbf{w}_1, mathbf{w}_2} is an orthogonal basis of a subspace W = Span(mathbf{w}_1, mathbf{w}_2) of mathbb{R}^3. Find a vector mathbf{n} which is orthogonal to W, and such that mathbf{v} - mathbf{n} is in W.

          10.12. Orthogonal component
Consider the following vectors:
mathbf{w}_1 = egin{bmatrix} -3 \ -4 \ 0 end{bmatrix}, quad mathbf{w}_2 = egin{bmatrix} 52 \ -39 \ -75 end{bmatrix}, quad mathbf{v} = egin{bmatrix} 0 \ -1 \ 1 end{bmatrix}
The set mathcal{B} = {mathbf{w}_1, mathbf{w}_2} is an orthogonal basis of a subspace W = Span(mathbf{w}_1, mathbf{w}_2) of mathbb{R}^3. Find a vector mathbf{n} which is orthogonal to W, and such that mathbf{v} - mathbf{n} is in W.

10.12. Orthogonal component
Consider the following vectors:
mathbfw1 = eginbmatrix -3 -4 0 endbmatrix, quad mathbfw2 = eginbmatrix 52 -39 -75 endbmatrix, quad mathbfv = eginbmatrix 0 -1 1 endbmatrix
The set mathcalB = mathbfw1, mathbfw2 is an orthogonal basis of a subspace W = Span(mathbfw1, mathbfw2) of mathbbR^3. Find a vector mathbfn which is orthogonal to W, and such that mathbfv - mathbfn is in W.

Added by Jackie C.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

05/19/2023

Step 1

To do this, we can take the cross product of W1 and W2. W1 = [5, 2, -3] W2 = [39, -75, 52] n = W1 x W2 = [(2 * 52) - (-3 * -75), (-3 * 39) - (5 * 52), (5 * -75) - (2 * 39)] Show more…

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10.12 Orthogonal Component Consider the following vectors: W1 = (3, 5, 2) W2 = (39, -75) The set B = {W1, W2} is an orthogonal basis of a subspace W = Span(W1, W2) of R3. Find a vector n which is orthogonal to W and such that V n is in W.