Question

Please help with this vectors question



Write a vector equation and a scalar equation of the plane.

Perpendicular to the line
[x, y, z] = [2, 4, -9] + t[3, 5, -3]
and including the point (4, -2, 7)

Answer from the book:
[x, y, z] = [4, -2, 7] + s[1, 0, 1] + t[0, 3, 5]
3x + 5y - 3z + 19 = 0

Please help me to arrive at that answer.

Answer 1

You would have to show at least some little work.

Answer 2

Can you please show me what work I would have to show?

Answer 3

I can show you some it.

Let's start with getting the scalar equation of the plane. You know that the plane you're looking for is of the form \[3x+5y-3z+d=0\]since you want a plane perpendicular to the line given to you. Now we just solve for d using the point (4, -2, 7). Plug in these values for \(x, y, z\), you get \[3(4)+5(-2)-3(7)+d=12-10-21+d=-19+d=0\]Hence, \(d=0\) and your scalar equation is\[3x+5y-3x+19=0\]

Answer 4

*Hence \(d=19\)

Answer 5

Oh, I see

Answer 6

How about for the other one?

Answer 7

For the other one, there are a few ways to do it. First off (this is easiest, and you'll get a different answer than what's in the book), pick 2 points in your plane other than \(p_0= (4, -2, 7)\). Call them \(p_1\) and \(p_2\). Now find the vector from \(p_0\) to \(p_1\) and from \(p_0\) to \(p_2\). The first vector will be multiplied by s, and the other by t. You'll get something like\[\left(\begin{matrix}4\\-2\\7\end{matrix}\right)+s\cdot\vec{p_0p_1}+t\cdot \vec{p_0p_2}\]

Answer 8

So can you show an example, since its going to be different from the book anyways

Answer 9

Lets say you chose the points (1, -5, -1) and (0, 1, 8). Then your vectors would be\[\left(\begin{matrix}-3\\-3\\8\end{matrix}\right)\]and\[\left(\begin{matrix}-4\\3\\-1\end{matrix}\right)\]Just plug these in for \(\vec{p_0p_1}\) and \(\vec{p_0p_2}\).

Answer 10

Sorry, the first vector should be \[\left(\begin{matrix}-3\\-3\\-8\end{matrix}\right)\]

Answer 11

How did you get those?

Answer 12

And the second should be\[\left(\begin{matrix}-4\\3\\1\end{matrix}\right)\]

Answer 13

Oops nevermind :P

Answer 14

How do you plug those in for PoP1 and PoP2?

Answer 15

The obvious way. You just plug them in. \[\left(\begin{matrix}4\\-2\\7\end{matrix}\right)+s\cdot\left(\begin{matrix}-3\\-3\\-8\end{matrix}\right)+t\cdot \left(\begin{matrix}-4\\3\\1\end{matrix}\right)\]

Answer 16

Is that the final answer?

Answer 17

This is a vector equation of the plane you are looking for. It is not the one your book chose. The book found an equation in a different manner, but yes, this is a vector equation of the plane.

Answer 18

Ok, THANKS SO MUCH! :)
By the way Im going to post 2 more questions similar to this one if you would like to help.