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Question
A mobile tower is situated at the top of a hill. Consider the surface on which the tower stands as a plane having points A(1, 0, 2), B(3, –1, 1) and C(1, 2, 1) on it. The mobile tower is tied with three cables from the points A, B and C such that it stands vertically on the ground. The top of the tower is at point P(2, 3, 1) as shown in the figure below. The foot of the perpendicular from the point P on the plane is at the point `Q(43/29, 77/29, 9/29)`.
Answer the following questions.
- Find the equation of the plane containing the points A, B and C.
- Find the equation of the line PQ.
- Calculate the height of the tower.
Solution
i. A(1, 0, 2), B(3, –1, 1), C(1, 2, 1)
Equation of plane
`|(x - x_1, y - y_1, z - z_1),(x_2 - x_1, y_2 - y_1, z_2 - z_1),(x_3 - x_1, y_3 - y_1, z_3 - z_1)| = 0`
`\implies |(x - 1, y - 0, z - 2),(3 - 1, -1 - 0, 1 - 2),(1 - 1, 2 - 0, 1 - 2)| = 0`
`\implies |(x - 1, y - 0, z - 2),(2, -1, -1),(0, 2, -1)| = 0`
`\implies` (x – 1)(1 + 2) – y(–2 – 0) + (z – 2)(4 – 0) = 0
`\implies` 3(x – 1) + 2y + 4(z – 2) = 0
`\implies` 3x – 3 + 2y + 4z – 8 = 0
`\implies` 3x + 2y + 4z = 11
ii. Let a and b be the position vectors of the points P(2, 3, 1) and `Q(43/29, 77/29, 9/29)` respectively.
Then, `veca = 2hati + 3hatj + hatk` and `vecb = 43/29hati + 77/29hatj + 9/29hatk`
Let `vecr` represent the position vector of any point A(x, y, z) on the line connecting points P and Q. The vector equation for the line is
`vecr = veca + λ(vecb - veca)`
= `(2hati + 3hatj + hatk) + λ[(43/29 - 2)hati + (77/29 - 3)hatj + (9/29 - 1)hatk]`
= `(2hati + 3hatj + hatk) + λ[((-15)/29)hati + ((-10)/29)hatj + ((-20)/29)hatk]`
Where λ is a parameter.
iii. The coordinates of the point P(2, 3, 1) and the equation of the plane in which the tower's bottom is located are 3x + 2y + 4z = 11.
Height of tower = `|(3(2) + 2(3) + 4(1) - 11)/sqrt(3^2 + 2^2 + 4^2)|`
= `|(6 + 6 + 4 - 11)/sqrt(9 + 4 + 16)|`
= `5/sqrt(29)`
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