Vector Equation (Dot Product?) with Parametrics

[FragKrag]

As part of a practice test, my Precalculus teacher assigned us a few problems. I can get most of them, but there are two I can't seem to figure out. They essentially the same problem, so I'll just post one.

1. The line whose vector equation is: \(\displaystyle A(t) = A_{1}+A_{2}+A_{3} = (A_{1},B_{2},C_{3}) = (13,11,74)+t(13,25,-22)\) is parallel to the plane...
Note: A(t) is a vector with component vectors \(\displaystyle A_{1}+A_{2}+A_{3}\)

a. 3x+2y+8z=6
b. x-y+3z=89
c. 2x+2y+z=103
d. 2x+6y+8z=1
e. 9x+5y-z=3

To be completely honest, I have no idea how to begin the problem, but I don't there are many thoughts to it. I think I'm missing something basic, but I can't really pinpoint it.

Thanks for any help in advance!

 

[Deleted member 4993]

As part of a practice test, my Precalculus teacher assigned us a few problems. I can get most of them, but there are two I can't seem to figure out. They essentially the same problem, so I'll just post one.

1. The line whose vector equation is: \(\displaystyle A(t) = A_{1}+A_{2}+A_{3} = (A_{1},B_{2},C_{3}) = (13,11,74)+t(13,25,-22)\) is parallel to the plane...
Note: A(t) is a vector with component vectors \(\displaystyle A_{1}+A_{2}+A_{3}\)

a. 3x+2y+8z=6
b. x-y+3z=89
c. 2x+2y+z=103
d. 2x+6y+8z=1
e. 9x+5y-z=3

To be completely honest, I have no idea how to begin the problem, but I don't there are many thoughts to it. I think I'm missing something basic, but I can't really pinpoint it.

Thanks for any help in advance!

Find a unit vector along the given line.

Find normals (vectors) to the planes given in answer choiceces.

Then find dot product of the unit vector with each of the normal vectors.

and continue....

 

[galactus]

The normal of the plane, \(\displaystyle ax+by+cz+d=0\) has normal \(\displaystyle n=[a,b,c]\).

I assume you know the normal is perpendicular to the plane. If it is perpendicular to the plane, then it must be parallel to the line.

Since you are given the choices, make it easy and try the dot products until you get 0.

The dot product is 0 if they are perpendicular.

For instance, the first choice: \(\displaystyle DotP([13,25,-22]\cdot [3,2,8])=-87\)

Not parallel.

 

[FragKrag]

Thanks! I think the (13,11,74) threw me off

I didn't think it'd be that simple though