Find the value of k so that the lines (x - 3)/(3k + 1) = (y + 6)/2 = (z + 3)/2k and quantity (x + 7)/3 = (y + 8)/-2k = (z + 9)/-3 are perpendicular.
the dot product must be zero. That is,
(3k+1)(3) + 2(-2k) + (2k)(-3) = 0
k = 3
To determine the value of k for which the two lines are perpendicular, we can use the property that the direction ratios of two perpendicular lines are negative reciprocals of each other.
Let's first find the direction ratios of both lines.
1. Line 1: (x - 3)/(3k + 1) = (y + 6)/2 = (z + 3)/(2k)
Comparing the ratios, we can write:
(x - 3)/(3k + 1) = (y + 6)/2 = (z + 3)/(2k) = A (say)
We will find the direction ratios using the values of A.
Direction ratios for Line 1:
a₁ = coefficient of x in A = 1
a₂ = coefficient of y in A = 1/2
a₃ = coefficient of z in A = 1/(2k)
2. Line 2: (x + 7)/3 = (y + 8)/(-2k) = (z + 9)/(-3)
Comparing the ratios, we can write:
(x + 7)/3 = (y + 8)/(-2k) = (z + 9)/(-3) = B (say)
We will find the direction ratios using the values of B.
Direction ratios for Line 2:
b₁ = coefficient of x in B = 1
b₂ = coefficient of y in B = 1/(-2k)
b₃ = coefficient of z in B = 1/(-3)
Now, according to the property, the direction ratios of two perpendicular lines are negative reciprocals of each other. Therefore, we have the following relationship:
a₁/b₁ = -1/a₂b₂ = -1/a₃b₃
1/(1*(-2k)) = - 1/(1/2*(-3k))
-2k = 6k
Solving this equation:
8k = 0
k = 0
Therefore, the value of k for which the lines are perpendicular is k = 0.