1. ## 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.

2. 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

3. Hello, Jenny!

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

4. Thank you very much Soroban. Your explanation is very clear to me.

Also, Thank you very much CaptainBlack.