Q. 233.7( 3 Votes )

Find the coordinates 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).

Answer :

Given: perpendicular drawn from point A (1, 8, 4) to line joining points B (0, -1, 3) and C (2, -3, -1)


To find: foot of perpendicular


Formula Used: Equation of a line is


Vector form:


Cartesian form:


where is a point on the line and with b1 : b2 : b3 being the direction ratios of the line.


If 2 lines of direction ratios a1:a2:a3 and b1:b2:b3 are perpendicular, then a1b1+a2b2+a3b3 = 0


Explanation:


B (0, -1, 3) is a point on the line.


Therefore,


Also direction ratios of the line are (0 - 2) : (-1 + 3) : (3 + 1)


-2 : 2 : 4


-1 : 1 : 2


So, equation of the line in Cartesian form is



Any point on the line will be of the form (-λ, λ - 1, 2λ + 3)


So the foot of the perpendicular is of the form (-λ, λ - 1, 2λ + 3)


The direction ratios of the perpendicular is


(-λ - 1) : (λ – 1 - 8) : (2λ + 3 - 4)


(-λ - 1) : (λ – 9) : (2λ – 1)


From the direction ratio of the line and the direction ratio of its perpendicular, we have


-1(-λ - 1) + λ – 9 + 2(2λ – 1) = 0


λ + 1 + λ – 9 + 4λ – 2 = 0


6λ = 10



So, the foot of the perpendicular is


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