find the projection of vector B=2i-4j-2k in the direction of vector A=2i-8j+k
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The projection of A onto B, denoted by Proj_{B} A, is a vector in the direction of B and with the magnitude of ||A|| (cos theta), where theta is the angle between the two vectors. So,
Proj_{B} A = ||A|| (cos theta) B_{u}, —— > (1)
where B_{u} is the unit vector in the direction of B, that is,
B_{u} = (1/||B||) B. From B = 3 i - 4 j - 12 k, we get
B_{u} = (1/13) [3i - 4j - 12k]. ——- > (2)
From the definition of the dot product and the Cartesian representation of A and B, we have
||A|| cos theta = (A.B)/||B|| = 26/13 = 2. —— > (3)
From (1)-(3), we obtain:
Proj_{B} A = (2/13) [3i - 4j - 12k].
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