The position vector of a point P relative to origin O is r = 3i + 4j + 5k. What is the magnitude of the projection of r onto the line whose direction vector is 2i + 3j + 6k?

Why is this the answer?

The projection of vector r onto a direction vector v is given by |r · u|, where u is the unit vector of v. v = 2i + 3j + 6k. |v| = sqrt(2² + 3² + 6²) = sqrt(4 + 9 + 36) = sqrt(49) = 7. u = (2/7)i + (3/7)j + (6/7)k. r · u = (3*2/7) + (4*3/7) + (5*6/7) = (6 + 12 + 30) / 7 = 48 / 7 ≈ 6.857. The closest option is 6.86, but given the options, there may be a typo in the question or options. Re-checking: 48/7 is 6.86. If the projection is onto the vector, the magnitude is 6.86.

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