# planes in 3-space

• Dec 12th 2006, 04:22 AM
Jenny20
planes in 3-space
Question
Find an equation for the plane passing through the point (0,4,1) which is parallel to the vectors <1,2,3> and <0,-1,1>.

Could you please teach me do this question? Thank you very much.
• Dec 12th 2006, 04:42 AM
CaptainBlack
Quote:

Originally Posted by Jenny20
Question
Find an equation for the plane passing through the point (0,4,1) which is parallel to the vectors <1,2,3> and <0,-1,1>.

Could you please teach me do this question? Thank you very much.

Take:

P(lambda,mu)=(0,4,1)+lambda(1,2,3)+mu(0,-1,1)

Then P(lambda,mu) is a point on a plane which passess through (0,4,1), and
and the plane is parallel to the plane defined by (0,0,0), (1,2,3), (0,-1,1).

RonL
• Dec 12th 2006, 05:01 AM
Soroban
Hello, Jenny!

Quote:

Find an equation for the plane passing through the point P(0,4,1)
which is parallel to the vectors u = <1,2,3> and v = <0,-1,1>.

Since the plane is parallel to u and v,
. . its normal vector n is perpendicular to both u and v.
Then n is the cross product of u and v.
Code:

```        | i  j  k | n  =  | 1  2  3 |  =  5i - j - k  =  <5,-1,-1>         | 0 -1  1 |```

You have a point P(0,4,1) and normal vector n = <5,-1,-1>

You can write the equation: .5(x - 0) - 1(y - 4) - 1(z - 1) .= .0

• Dec 12th 2006, 06:13 PM
Jenny20
Thank you very much Soroban. Your explanation is very clear to me. :)

Also, Thank you very much CaptainBlack.