Q. 164.7( 7 Votes )
Find the length and the foot of the perpendicular drawn from the point (2, -1, 5) to the line
Given: Equation of line is
To find: coordinates of foot of the perpendicular from (2, -1, 5) to the line. And find the length of the perpendicular.
1. Equation of a line is
where is a point on the line and b1 : b2 : b3 is the direction ratios of the line.
2. Distance between two points (x1, y1, z1) and (x2, y2, z2) is
So the foot of the perpendicular is (10λ + 11, -4λ - 2, -11λ - 8)
Direction ratio of the line is 10 : -4 : -11
Direction ratio of the perpendicular is
⇒ (10λ + 11 - 2) : (-4λ - 2 + 1) : (-11λ - 8 - 5)
⇒ (10λ + 9) : (-4λ - 1) : (-11λ - 13)
Since this is perpendicular to the line,
10(10λ + 9) - 4(-4λ - 1) - 11(-11λ - 13) = 0
⇒ 100λ + 90 + 16λ + 4 + 121λ + 143 = 0
⇒ 237λ = -237
⇒ λ = -1
So the foot of the perpendicular is (1, 2, 3)
= √14 units
Therefore, the foot of the perpendicular is (1, 2, 3) and length of perpendicular is √14 units.
Rate this question :
Find the equation of the plane through the line of intersection of and perpendicular to the plane Hence find whether the plane thus obtained contains the line
Find the vector and Cartesian equations of a line passing through (1, 2, -4) and perpendicular to the two lines andMathematics - Board Papers
Find the vector and Cartesian equations of the line through the point (1, 2, -4) and perpendicular to the two lines.
andMathematics - Board Papers
Write the vector equations of the following lines and hence determine the distance between them:
Mathematics - Board Papers
Find the value of p, so that the lines and are perpendicular to each other. Also find the equations of a line passing through a point (3, 2, –4) and parallel to line l1.Mathematics - Board Papers