Question

Projection of a line segment joining the points (2,0,5) and (0, 3, 1) on the line whose direction ratios are 2, 3, 6 is ____

Answer 1

The projection of the line segment joining the points (2,0,5) and (0, 3, 1) onto the line with direction ratios 2, 3, 6 is -19/7.

Step-by-step explanation:

To find the projection of the line segment joining the points (2,0,5) and (0, 3, 1) onto the line whose direction ratios are 2, 3, 6, we use the following procedure:

1. First, find the vector representing the line segment by subtracting the coordinates of the two points, AB = B - A = (0 - 2, 3 - 0, 1 - 5) = (-2, 3, -4).
2. The direction ratios of the given line can be taken as the components of a direction vector, so let's denote this vector as d = (2, 3, 6).
3. To find the projection of vector AB onto vector d, we use the formula for the scalar projection: projd(AB) = (AB · d) / |d|, where · denotes the dot product and |d| is the magnitude of vector d.
4. Calculate the dot product AB · d = (-2*2) + (3*3) + (-4*6) = -4 + 9 - 24 = -19.
5. Calculate the magnitude of vector d, |d| = √(2^2 + 3^2 + 6^2) = √(4 + 9 + 36) = √49 = 7.
6. Finally, compute the projection using the scalar projection formula: projd(AB) = -19 / 7.

Hence, the projection of the line segment on the given line is -19/7.