find the foot of the perpendicular from P(7,-3,1) to the plane 4x-8y+z=-1.?

Question

In other words, find the distant from P to the plane.

Answer 1

Find the foot of the perpendicular from P(7,-3,1) to the plane 4x - 8y + z = -1. In other words, find the distance from P to the plane.

The question posed sounds different than the explanation given "in other words". It sounds like it is asking for the closest point on the plane to P. So I will calculate both.

First calculate the distance from P(7, -3, 1) to the plane.

Write the equation of the plane and set equal to zero.

4x - 8y + z + 1 = 0

Calculate the distance d from point P(7,-3,1) to the plane.

d = | 4*7 - 8*(-3) + 1*1 + 1 | / √(4² + (-8)² + 1²)

d = | 28 + 24 + 1 + 1| / √(16 + 64 + 1)

d = 54 / √81 = 54 / 9 = 6
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Now calculate the foot of the perpendicular from P(7,-3,1) to the plane. In other words calculate the closest point on the plane to P.

Write the normal line r(t) to the plane thru P(7, -3, 1).

r(t) = OP + tn = <7, -3, 1> + t<4, -8, 1>
r(t) = <7 + 4t, -3 - 8t, 1 + t>
where t is a scalar that ranges over the real numbers

Rewrite the equation of the plane in terms of the line and solve for t.

4x - 8y + z + 1 = 0

4(7 + 4t) - 8(-3 - 8t) + 1(1 + t) + 1 = 0
28 + 16t + 24 + 64t + 1 + t + 1 = 0
81t + 54 = 0
t = -54/81 = -2/3

x = 7 + 4t = 7 + 4(-2/3) = 13/3
y = - 3 - 8t = - 3 - 8(-2/3) = 7/3
z = 1 + t = 1 - 2/3 = 1/3

The foot of the perpendicular is (13/3, 7/3, 1/3).

Answer 2

you can try elimination method -2x - 4y + z = -2 equation 1 3x - y + z = 1 equation 2 4x + 8y -2z = 5 equation 3 multiply 1 and 2 w/ something that will make a variable into 0 3[ -2x -4y +z = -2] 2 [ 3x - y + z = 1] then add the two equation (-6x - 12y + 3z = -6) +(6x - 2 y + 2z = 2) ---------------------- -14y + 5z = 4 (you get what just happened here?) now you got equation 4 do the same with 2 and 3 since if you do it at 1 and 3 everything will become 0 4[3x - y + z = 1] -3[4x + 8y -2z = 5] you'll get 12x - 4y + 4z = 4 -12x -24y +6z = -15 ---------------------- 28y +10z = -11 and this is equation 5 combine 4 and 5 you'll get -14y + 5z = 4 28y +10z = -11 by now you get the pattern so ill speed up 2[-14y + 5z = 4] 1[28y +10z = -11] --------------------------------- +20z = -3 z = -3/20 substitute with equation 4 or 5 -14y + 5(-3/20) = 10 -14y = 10 + 1/4 -14y = 41/4 y = -41/56 please check i dont have my scientific calculator with me then substitute z and y withe either equation 1 2 or 3 ill go for the easiest 3x - y + z = 1 3x + 41/56 - 3/20 = 1 oh crap you have a calculator you finish it its your destiny.... hahaha

Answer 3

1) 4X-8Y+Z = -1
4(7)-8(-3)+Z = -1
28 +24+Z = -1
52 + Z = -1
Z = -1-52
Z = -51

2) 4X-8Y+Z = -1
4X-8(-3)+(1) = -1
4X+24+1 = -1
4X+25 = -1
4X = -25-1
4X = -24
X = -24/4
X = -6

3) 4X-8Y+Z = -1
4(7)-8Y+(1) = -1
28-8Y = -1
-8Y = -1-28
-8Y = -29
Y = 29/8