Question

Find a vector equation and parametric equations for the line.

The line through the point (1, 0, 6) and perpendicular to the plane x+3y+z=5

Answer 1

the two vectors are parallel

Answer 2

okay can you prove it mathematically?

Answer 3

there is some scalar t such that \[<1,3,1> = t<x-1,y-0,z-6>\]

Answer 4

calculus are you still typing?

Answer 5

@raffle_snaffle what @sirm3d is trying to tell you is that a normal vector is a vector that is perpendicular to the plane of a vector and parallel to any vector, perpendicular to the plane. What that means is simply, if a line passes through a point and is perpendicular to a given plane, then the line is parallel to the normal of the plane. If the equation of a plane is\[Ax +By +Cz =D\]then the normal vector to the plane is\[<A, B, C > \]and if there is a line perpendicular to this plane, then the line is parallel to the normal of the plane.

Now a vector equation of a line is given by\[<x,y,z > =<x _{0},y _{0},z _{0}>+t d; t \in \mathbb{R} \]where t is a parameter (any real number) and d is a direction vector of the line.

Thus suppose, as per your problem, if we have a line passing through the point\[(x _{0},y _{0},z _{0})\]and perpendicular to the plane\[Ax +By +Cz =D \]then the vector equation of this line with the direction vector\[<A,B,C > \]would be\[<x,y,z >=<x _{0},y _{0},z _{0}>+t <A,B,C >; t \in \mathbb{R} \]

Answer 6

i understand what you are saying up top

Answer 7

Well @raffle_snaffle ? Do you see the visual @sirm3d drew, coupled with my explanation? If you need further clarification, just ask.

Answer 8

another pic would be nice.

Answer 9

I understand a normal vector is a vector the is perpendicular to all the vectors that are within the plane

Answer 10

Sorry, I don't think I can draw that as well, in this format. If @sirm3d or someone else would like to attempt another drawing. If have specific questions about the lesson, then I'll be happy to do so.

Answer 11

Yes

Answer 12

how do we solve this problem

Answer 13

so the vectors in the plane is <x-1, y-0, z-6>

Answer 14

What problem? Your problem? It's exactly the same as my lesson. I'm here to teach so you do the question and then I will correct it, but I'm not going to do it for you.

Answer 15

Yes

Answer 16

applying the dot product to n and v will result in 0, right?

Answer 17

What do you think, is the vector equation for your problem?

Answer 18

oh... yes

Answer 19

OK so what is the equation?

Answer 20

well i need to still obtain the normal vector have not done it yet

Answer 21

Look at your problem carefully. The normal vector is already given. Nothing to find.

Answer 22

i am working on it

Answer 23

@sirm3d showed you what the normal vector is in his diagram, and I told you what the normal vector is in the lesson. Now put two and two together. No worries. Take your time.

Answer 24

okay i got x+3y+z=7

Answer 25

Yes?

Answer 26

and it checks if we plug the given point

Answer 27

that is the equation of the plane

Answer 28

OK but what is the vector equation, you're seeking?