The projection of a onto b for a = (4,−2,2), b = (1,0,-1) is which vector?

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Multiple Choice

The projection of a onto b for a = (4,−2,2), b = (1,0,-1) is which vector?

The projection of a onto b is the component of a in the direction of b, given by ((a·b)/(b·b)) times b. Compute the dot products: a·b = 4·1 + (-2)·0 + 2·(-1) = 2, and b·b = 1^2 + 0^2 + (-1)^2 = 2. The scalar factor is 2/2 = 1, so the projection is 1·b = (1, 0, -1). Since the factor is 1, the projection points in the same direction as b and has the same length as b, hence it equals (1, 0, -1). The other options would correspond to a different scalar multiple of b or a vector not parallel to b, which cannot be the projection.

The projection of a onto b is the component of a in the direction of b, given by ((a·b)/(b·b)) times b. Compute the dot products: a·b = 4·1 + (-2)·0 + 2·(-1) = 2, and b·b = 1^2 + 0^2 + (-1)^2 = 2. The scalar factor is 2/2 = 1, so the projection is 1·b = (1, 0, -1). Since the factor is 1, the projection points in the same direction as b and has the same length as b, hence it equals (1, 0, -1). The other options would correspond to a different scalar multiple of b or a vector not parallel to b, which cannot be the projection.

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