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

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Answer 2

The equation of the projection of L on m is:

x + 8/7 = (y - 9/7)/3 = (z + 1/7)/6

Given: The direction ratios of the line are 2, 3, and 6.  

To Find: The equation of the projection of L on m

Solution: Let the line segment joining the points (2,0,5) and (0,3,1) be denoted by L.

• The direction ratios of the given line are 2, 3, and 6. Let this line be denoted by m.
• To find the projection of L on m, we need to find the point on m that is closest to L. This point is the intersection of m and the perpendicular from L to m.
• Let P (x, y, z) be the point on m that is closest to L. Then the vector L - P is perpendicular to m.

The vector L - P is given by:

L - P = (2 - x, 3 - y, 1 - z)

Since L - P is perpendicular to m, we have:

(2 - x)/2 = (3 - y)/3 = (1 - z)/6

Solving these equations, we get:

x = -8/7, y = 9/7, z = -1/7

• Therefore, the point on m closest to L is P(-8/7, 9/7, -1/7).
• The projection of L on m is the line passing through P and perpendicular to m.
• The direction ratios of this line are the same as those of m, i.e., 2, 3, 6.

Therefore, the equation of the projection of L on m is:

x + 8/7 = (y - 9/7)/3 = (z + 1/7)/6.

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