The projection of the line segment joining the points (1, –1, 3) and (2, –4, 11) on the line joining the points (–1, 2, 3) and (3, –2, 10) is ____________.

Solution

Maths sir · 3 steps

Turn both pairs of points into vectors, then dot the segment with the unit vector along the line.

1. Segment vector. From (1, -1, 3) to (2, -4, 11) the segment is a⃗ = (1, -3, 8).
2. Line direction. From (-1, 2, 3) to (3, -2, 10) the direction is b⃗ = (4, -4, 7), with |b⃗| = √(16 + 16 + 49) = √81 = 9.

   Projection = (a⃗·b⃗)/|b⃗| = ((1)(4) + (-3)(-4) + (8)(7))/9 = (4 + 12 + 56)/9 = 72/9

3. So. The projection is 72/9 = 8.

The common slip: 72. Stopping at the dot product and forgetting to divide by |b⃗| = 9 gives 72, not 8.

The idea to keep. Projection of a segment on a line is the dot product of the segment vector with the line's unit vector. Divide by the length of the line you project onto, never by the segment's own length.

Answer from the NTA final answer key. Solution by PrepWiser.