Question

Find the co-ordinates of the foot of the perpendicular drawn from the point A(1, 8, 4) to the line joining the points B (0, –1, 3) and C (2, – 3, – 1).

Solution

The equation of line through B (0, –1,3) and C (2, –3, – 1) is

fraction numerator straight x minus 0 over denominator 2 minus 0 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 3 plus 1 end fraction space equals space fraction numerator straight z minus 3 over denominator negative 1 minus 3 end fraction

straight x over 2 space equals space fraction numerator straight y plus 1 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 3 over denominator negative 4 end fraction

straight x over 1 space equals space fraction numerator straight y plus 1 over denominator negative 1 end fraction space equals space fraction numerator straight z minus 3 over denominator negative 2 end fraction space equals space straight k comma space space space say

Any point D on its (k, – k – 1, –2, k + 3)

Let D be foot of perpendicular form A on BC.
Direction ratios of AD are
k – 1, – k – 1 – 8, – 2 k + 3 – 4
i.e. k – 1, – k – 9, –2 k – 1
Direction ratios of BC are
2 – 0, – 3 + 1, – 1 – 3 i.e. 2, –2, – 4
i.e. 1, –1, –2.
Since AD is perpendicular to BC
∴ (k – 1) (1) + (– k – 9) (–1) + (–2 k – 1) (– 2) = 0
$therefore space space straight k minus 1 space plus space straight k space plus 9 space plus space 4 space straight k space plus space 2 space equals space 0 space space space space rightwards double arrow space space 6 space straight k space equals space minus 10 space space rightwards double arrow space space straight k space equals space minus 5 over 3 therefore space space space foot space of space perpendicular space straight D space is space open parentheses negative 5 over 3 comma space 5 over 3 comma space minus 1 comma space 10 over 3 plus 3 close parentheses space space straight i. straight e. space open parentheses negative 5 over 3 comma space 2 over 3 comma space 19 over 3 close parentheses$

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Three Dimensional Geometry

Find the co-ordinates of the foot of the perpendicular drawn from the point A(1, 8, 4) to the line joining the points B (0, –1, 3) and C (2, – 3, – 1).

Some More Questions From Three Dimensional Geometry Chapter

If α, β, γ are the angles which a line makes with the axes, prove that sin²α + sin² β + sin²γ = 2.

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NCERT Sample Papers

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For Daily Free Study Material Join wiredfaculty

Three Dimensional Geometry

Find the co-ordinates of the foot of the perpendicular drawn from the point A(1, 8, 4) to the line joining the points B (0, –1, 3) and C (2, – 3, – 1).

Some More Questions From Three Dimensional Geometry Chapter

If α, β, γ are the angles which a line makes with the axes, prove that sin2α + sin2 β + sin2γ = 2.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

If α, β, γ are the angles which a line makes with the axes, prove that sin²α + sin² β + sin²γ = 2.