Compute cos θ where θ is the angle between a=(1,2,3) and b=(-1,0,1). Which of the following equals cos θ?

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Multiple Choice

Compute cos θ where θ is the angle between a=(1,2,3) and b=(-1,0,1). Which of the following equals cos θ?

Cosine of the angle between two vectors comes from the dot product: cos theta = (a · b) / (||a|| ||b||). Compute the dot product: a · b = 1(-1) + 2(0) + 3(1) = -1 + 0 + 3 = 2. Find the magnitudes: ||a|| = sqrt(1^2 + 2^2 + 3^2) = sqrt(14), and ||b|| = sqrt((-1)^2 + 0^2 + 1^2) = sqrt(2). Thus cos theta = 2 / (sqrt(14) * sqrt(2)) = 2 / sqrt(28) = 2 / (2 sqrt(7)) = 1 / sqrt(7). Therefore, cos theta = 1/√7.

Cosine of the angle between two vectors comes from the dot product: cos theta = (a · b) / (||a|| ||b||).

Compute the dot product: a · b = 1(-1) + 2(0) + 3(1) = -1 + 0 + 3 = 2.

Find the magnitudes: ||a|| = sqrt(1^2 + 2^2 + 3^2) = sqrt(14), and ||b|| = sqrt((-1)^2 + 0^2 + 1^2) = sqrt(2).

Thus cos theta = 2 / (sqrt(14) * sqrt(2)) = 2 / sqrt(28) = 2 / (2 sqrt(7)) = 1 / sqrt(7).

Therefore, cos theta = 1/√7.

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