Find the value of p for which the line through the points A(3, 5, -1)

1.	Find the value of p for which the line through the points A(3, 5, -1) and B(5, p, 0) 9 is perpendicular to the line through the points C(2, 1, 1) and D (3, 3, -1).
Answer» Given: A(3, 5, -1) and B(5, p, 0) The line joining these two points is given by, AB = 2i + (p-5)j + k C(2, 1, 1) and D(3, 3, -1), The line joining these two points, CD = i + 2j - 2k As the two lines are perpendicular, we know that the angle between these two lines is So, AB.CD = 0 (dot product) Thus, (2i + (p-5)j + k) .(i + 2j - 2k) = 0. 2 + 2(p - 5) - 2 = 0 p = 5 Thus, p = 5.

Find the value of p for which the line through the points A(3, 5, -1) and B(5, p, 0) 9 is perpendicular to the line through the points C(2, 1, 1) and D (3, 3, -1).

Answer»

Given: A(3, 5, -1) and B(5, p, 0)

The line joining these two points is given by,

AB = 2i + (p-5)j + k

C(2, 1, 1) and D(3, 3, -1),

The line joining these two points,

CD = i + 2j - 2k

As the two lines are perpendicular, we know that the angle between these two lines is

So, AB.CD = 0 (dot product)

Thus, (2i + (p-5)j + k) .(i + 2j - 2k) = 0.

2 + 2(p - 5) - 2 = 0

p = 5

Thus, p = 5.