Find the equation for the plane containing the point

[number0]

Homework Statement

Find the equation for the plane containing the point (3,-8,7) and the line

l(t)=<1,-8,5>+t<3,-2,5>

Homework Equations

Cross product

Equation of a plane: A(x-x₀)+B(y-y₀)+C(z-z₀)+D=0

The Attempt at a Solution

First I set t=0 in l(t)=<1,-8,5>+t<3,-2,5> to obtain the point (1,-8,5)

I then proceed to to subtract the given point (3,-8,7) by (1,-8,5) to get a vector <2,0,2>.

Since I have two vectors, that is, <3,-2,5> and <2,0,2>, I can use the cross product on

the two vectors to obtain the vector for the plane. By <3,-2,5> x <2,0,2>, I obtained the

vector <-4,4,4>. Simplifying it would result in <-1,1,1>.

Thus, using the equation for the plane:

A(x-x₀)+B(y-y₀)+C(z-z₀)+D=0

where <A,B,C> = <-1,1,1> and (x₀,y₀,z₀) = (3,-8,7)

So now, the equation looks like this:

-(x-3)+(y+8)+(z-7)=0

-x+3+y+8+z-7=0

-x+y+z=-4I am not even sure if I did it right. This question was a question on my last exam (and the

only one that I missed). Sadly, my professor did not even post up the solutions. Can anyone

please see if my results and methods were correct? Thanks! :)

 

[Mark44]

You can check your answer yourself.
Does your plane contain the point (3, -8, 7)?
Is every point in the given line also on your plane?

If the answers to both questions are yes, then your work is correct.