Answer:
[tex]3x - 4y + z = 1[/tex]
Step-by-step explanation:
Given
[tex]Point\ 1 = (4,3,1)[/tex]
[tex]Point\ 2 = (4,1,-7)[/tex]
Perpendicular to [tex]8x + 7y + 4z = 18[/tex]
Required
Determine the plane equation
The general equation of a plane is:
[tex]a(x-x_1) + b(y - y_1) + c(z-z_1) = 0[/tex]
For [tex]n = <a,b,c>[/tex]
[tex](x_1,y_1,z_1) = (4,3,1)[/tex]
[tex](x_2,y_2,z_2) = (4,1,-7)[/tex]
First, we need to determine parallel vector [tex]V_1[/tex]
[tex]V_1 = <x_1 - x_2, y_1 - y_2, z_1 - z_2>[/tex]
[tex]V_1 = <4 - 4, 3 - 1, 1 - (-7)>[/tex]
[tex]V_1 = <0,2,8>[/tex]
[tex]V_1[/tex] is parallel to the required plane
From the question, the required plane is perpendicular to [tex]8x + 7y + 4z = 18[/tex]
Next, we determine vector [tex]V_2[/tex]
[tex]V_2 = <8,7,4>[/tex]
This implies that the required plane is parallel to [tex]V_2[/tex]
Hence: [tex]V_1[/tex] and [tex]V_2[/tex] are parallel.
So, we can calculate the cross product [tex]V_1 * V_2[/tex]
[tex]V_1 = <0,2,8>[/tex]
[tex]V_2 = <8,7,4>[/tex]
[tex]n = V_1 * V_2[/tex]
[tex]V_1 * V_2 =\left[\begin{array}{ccc}i&j&k\\0&2&8\\8&7&4\end{array}\right][/tex]
The product is always of the form + - +
So:
[tex]V_1 * V_2 = i\left[\begin{array}{cc}2&8\\7&4\end{array}\right][/tex] [tex]-j\left[\begin{array}{cc}0&8\\8&4\end{array}\right][/tex] [tex]+k\left[\begin{array}{cc}0&2\\8&7\end{array}\right][/tex]
Calculate the product
[tex]V_1 * V_2 = i(2*4- 8*7) - j(0*4- 8*8) + k(0*7 - 2 * 8)[/tex]
[tex]V_1 * V_2 = i(8- 56) - j(0- 64) + k(0 - 16)[/tex]
[tex]V_1 * V_2 = i(-48) - j(- 64) + k(- 16)[/tex]
[tex]V_1 * V_2 = -48i +64j - 16k[/tex]
So, the resulting vector, n is:
[tex]n = <-48,64,-16>[/tex]
Recall that:
[tex]n = <a,b,c>[/tex]
By comparison:
[tex]a = -48[/tex] [tex]b = 64[/tex] [tex]c = -16[/tex]
Substitute these values in [tex]a(x-x_1) + b(y - y_1) + c(z-z_1) = 0[/tex]
[tex]-48(x-x_1) + 64(y - y_1) -16(z-z_1) =0[/tex]
Recall that:[tex](x_1,y_1,z_1) = (4,3,1)[/tex]
So, we have:
[tex]-48(x-4) + 64(y - 3) -16(z-1) =0[/tex]
[tex]-48x + 192 + 64y -192 - 16z +16 = 0[/tex]
Collect Like Terms
[tex]-48x + 64y - 16z = 0 - 192 + 192 - 16[/tex]
[tex]-48x + 64y - 16z = -16[/tex]
Divide through by -16
[tex]3x - 4y + z = 1[/tex]
Hence, the equation of the plane is[tex]3x - 4y + z = 1[/tex]
