Question

Please help with this vectors question



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

Parallel to the yz-plane and including the point (-1, -2, 5).

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

Please show all work! Thanks :)

Answer 1

What's a vector that's perpendicular to the yz plane?

Answer 2

This is what the textbook question says word for word

Answer 3

This is what I got so far,
i=[1,0,0]
Ax+By+Cz+D=0
x+(0)y+(0)z+D=0
x+D=0
(-1)+D=0
D=1
scalar equation: x+1=0

Is this right?

Answer 4

That looks right to me. Now you just need to get the vector equation.

Answer 5

Yea, can you help me with that?

Answer 6

To do the vector equation, you need to find two vectors in that plane. This is surprisingly easy.

Answer 7

Please teach me because Im extremely confused...My friend showed me how to do it bu I dont understand, here Ill post it,

Answer 8

Its part b)

Answer 9

The vector that's perpendicular to the yz plane is just <1,0,0>, and there are 3 commonly used vectors that are all perpendicular to each other. These vectors are <1,0,0>, <0,1,0>, and <0,0,1>. If you're given one of these vectors, and asked for another perpendicular vector, choose another one of these.

Answer 10

Thus, you can use <0,1,0> and <0,0,1> for the two vectors needed in your vector equation.

As for the starting point, use (-1,2,5) since the plane must contain that line.

Answer 11

So, if I copy that sheet, is it correct?

Answer 12

Oh and by the way its [-1, -2, 5]

Answer 13

Almost. That sheet gives you one possible vector. You need to choose another value for y and z, and find a vector from (-1,-2,5) to that second point. Multiply this second vector by s, and add it to the equation.

Answer 14

Can you help me with that?

Answer 15

Pick another coordinate with x=1.

Answer 16

Im sorry lol but I dont know what you mean

Answer 17

(1, y, z) choose some values for y and z.

Answer 18

Any values?

Answer 19

any. although small integers are usually nice.

Answer 20

ok, how about [1, 3, 4]

Answer 21

Did I say x=1? I keep forgetting negative signs. It should be -1. So we'll go with (-1, 3, 4).

Answer 22

Ok

Answer 23

Now, what's the vector from (-1, -2, 5) to (-1, 3, 4)?

Answer 24

umm is it,
[-1 +1, 3+2, 4-5]
[0, 5, -1]
?

Answer 25

Bingo. So now you have the second vector you need.

Answer 26

Using the stuff written on the sheet, your vector equation would then be

<-1,-2,5>+t<0,2,-5>+s<0,5,-1>

Answer 27

Are you sure thats how it is?

Answer 28

Just like the other problem I did earlier, there are multiple vector equations for this one plane.

Answer 29

The "simplest" one is just the solution that the book gave you.

Answer 30

Ok, THANKS A TON!! :) <3
I have one more question similar to this if you'd like to help :P

Answer 31

@KingGeorge Is it like this?
[x, y, z]=[-1, -2, 5] + s[0, 5, -1] + t[0, 2, -5]

Answer 32

That is also correct.

Answer 33

Ok thanks!