Find the value of p for which the line through the points A (4, 1, 2)

1.	Find the value of p for which the line through the points A (4, 1, 2) and B (5, p, 0) is perpendicular to the line through the points C (2, 1, 1) and D (3, 3, -1).
Answer» It is given that Line joining A (4, 1, 2) and B (5, p, 0) is written as AB = i + (p – 1)j – 2k Line joining C (2, 1, 1) and D (3, 3, -1) is written as CD = i + 2j – 2k In order prove that these two lines are perpendicular we must show that angle between these two lines is π/2 Dot product AB . CD = 0 Substituting the values (i + (p – 1)j – 2k). (i + 2j – 2k) = 0 So we get 1 + 2 (p – 1) + 4 = 0 p = -3/2 Hence, value of p is -3/2.

Find the value of p for which the line through the points A (4, 1, 2) and B (5, p, 0) is perpendicular to the line through the points C (2, 1, 1) and D (3, 3, -1).

Answer»

It is given that

Line joining A (4, 1, 2) and B (5, p, 0) is written as

AB = i + (p – 1)j – 2k

Line joining C (2, 1, 1) and D (3, 3, -1) is written as

CD = i + 2j – 2k

In order prove that these two lines are perpendicular we must show that angle between these two lines is π/2

Dot product AB . CD = 0

Substituting the values

(i + (p – 1)j – 2k). (i + 2j – 2k) = 0

So we get

1 + 2 (p – 1) + 4 = 0

p = -3/2

Hence, value of p is -3/2.