Projection Of Vectors

Question

Consider the vectors a = 3i + 4j and b = i + 2j + 2k; find the scalar projection of vector a onto the direction of vector b.

RankGuru · Accepted Answer

The scalar projection of a onto b measures the signed length of the shadow of a along b and is given by (a . b)/|b|. This quantity, distinct from the vector projection, is a routine JEE Advanced computation linking dot product to length. First compute a . b = (3)(1) + (4)(2) + (0)(2) = 3 + 8 + 0 = 11, remembering that a has zero k-component. Next, |b| = sqrt(1^2 + 2^2 + 2^2) = sqrt(1 + 4 + 4) = sqrt(9) = 3. Therefore the scalar projection equals 11/3. Option 11/5 incorrectly uses |b| = 5, perhaps by summing squares without the root. Option 5 mistakes |b| for the magnitude of a instead. Option 11 forgets to divide by |b| altogether, reporting the bare dot product. The result 11/3 is positive, indicating a and b form an acute angle, consistent with the positive dot product found earlier. Plausibility check: the scalar projection cannot exceed |a| = sqrt(9 + 16) = 5, and 11/3 is about 3.67, which is indeed less than 5, so the magnitude constraint is respected.

Projection Of Vectors

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