Given: a point P (5, 4, 2), line .
To find: the length and coordinates of the foot of the perpendicular from the point P to the given line. Also find the image of point P in the given line.
Given line is
So the Cartesian form of the line will be,
Let Q(α, β, γ) be the foot of perpendicular drawn from P(5, 4, 2) to the line (i) and P’ (x1 , y1 , z1) be the image of P on the line (i)
Now as point Q lies on the given line, so let
So from above condition we get
Hence the coordinates of Q are (2k-1, 3k+3, 1-k)
Parallel vector of the given line is
Substituting corresponding values, we get
Now substituting the values of α, β, γ, we get
Substituting the value of k in coordinates of Q, we get
Q(2k-1, 3k+3, 1-k)
⇒ Q(2(1)-1, 3(1)+3, 1-(1))
⇒ Q(2-1, 3+3, 1-1)
⇒ Q(1, 6, 0)
Hence coordinates of the foot of perpendicular and the length of the perpendicular drawn from the point P (5, 4, 2) to the line is Q (1,6,0)
Now we will find the length of this perpendicular
Or the length of the perpendicular is √24 or 2√6 units.
Now Let P’(x1, y1, z1) be the image of point P (5, 4, 2) in the line
Now from figure it is clear that M(1, 6, 0) is the midpoint of line PP’
Hence now applying the formula
Hence the image of the given point P(5, 4, 2) is P’(-3, 8, -2).
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