Multivariable Problem :P

In summary, the line L through point Po(1,2,8) parallel to vector R(3i-j-4k) intersects the plane through point P1(-4,0,3) with normal vector n(3i-2j+6k) at the point (10,-1,-4). The equation of the plane is 3x-2y+6z-6=0. To find the equation of the plane, one can use the formula R.n=d where R is any point on the plane and n is the normal vector. By substituting the given point P1, the normal vector, and solving for d, we get the equation of the plane.
  • #1
multicalcprob
5
0
Let L be the line through the point Po(1,2,8) which is parallel to the vector R(3i-j-4k). Find the point at which l intersects the plane through the point p1(-4,0,3) having normal vector n(3i-2j+6k)

I did the following:
x=1+3t
y=2-t
z=8-4t

3(x+4)+6(z-3)=0
3x+6z-6=0

3(1+3t)+6(8-4t)-6=0
-15t+45=0
t=3

(10, -1, -4)

Did I do this right?
Thanks.
 
Last edited:
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  • #2
I got a different answer, I'm not quite sure how you're working out the equation of your plane. I ended up with the equation 3x-2y+6z-6=0, which doesn't give quite so nice an answer I'm afraid! Could you explain how your method for working out the equation of the plane works??

I did it like this:
equation of a plane is R.n=d
you are given n and you are given one specific R,
so equation is R.(3 -2 6) = d

now you substitute in the specific r to find the d, if that makes sense, so:

(-4 0 3).(3 -2 6) = -12 +0 +18 = 6

so you know the equation of the plane is
R.(3 -2 6) =6

or, in other words (taking R to be (x y z):
3x-2y+6z=6
3x-2y+6z-6=0
 
  • #3
multicalcprob said:
3(x+4)+6(z-3)=0
3x+6z-6=0

I believe you omitted the y-term in your plane equation.
Weatherhead's result for the plane looks right to me.
 

1. What is a multivariable problem?

A multivariable problem is a mathematical problem that involves more than one independent variable. This means that the outcome of the problem is affected by multiple factors and cannot be solved by considering only one variable.

2. How do you solve a multivariable problem?

To solve a multivariable problem, you can use various mathematical techniques such as substitution, elimination, or graphing. The specific method used will depend on the type of problem and the number of variables involved.

3. What are some common examples of multivariable problems?

Multivariable problems can be found in fields such as physics, economics, and engineering. Some common examples include optimization problems, systems of equations, and regression analysis.

4. What are some challenges of solving multivariable problems?

One of the main challenges of solving multivariable problems is that they can become increasingly complex as the number of variables increases. It can also be difficult to determine which variables are most important or have the greatest impact on the outcome.

5. How are multivariable problems used in real-world applications?

Multivariable problems are used in a wide range of real-world applications, from predicting stock market trends to designing efficient transportation systems. These problems allow scientists and researchers to analyze and understand complex systems and make data-driven decisions.

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