Does this line belong to this plane?

[pharoh123]

i am given an equation of a line: [x,y,z]=[-3,-6,-11]+k[22,1,-11]
and i am being asked if it belongs to the plane defined by these three points A(2,5,6) B(-7,1,4) and C(6,-2,-9)
so first i calculated all three vectors between the points so i have AB=[-9,-4,2] AC=[4,-7,-15] BC=[13,-3,-13] and clearly none are collinear with [22,-1,11] so clearly the line is not part of the plane yet the book claims that it is part of the plane. Am i doing something wrong?

 

[Mark44]

i am given an equation of a line: [x,y,z]=[-3,-6,-11]+k[22,1,-11]
and i am being asked if it belongs to the plane defined by these three points A(2,5,6) B(-7,1,4) and C(6,-2,-9)
so first i calculated all three vectors between the points so i have AB=[-9,-4,2] AC=[4,-7,-15] BC=[13,-3,-13] and clearly none are collinear with [22,-1,11] so clearly the line is not part of the plane yet the book claims that it is part of the plane. Am i doing something wrong?

Your given line can be in the plane without having to be collinear with any of the three lines you found. You can determine whether the given line is in the plane by finding the equation of the plane, and then determining whether the given line satisfies that plane equation.

Do you know how to find the equation of a plane given three points in the plane?

 

[pharoh123]

Your given line can be in the plane without having to be collinear with any of the three lines you found. You can determine whether the given line is in the plane by finding the equation of the plane, and then determining whether the given line satisfies that plane equation.

Do you know how to find the equation of a plane given three points in the plane?

yeah i could just put [x,y,z]=A + s*AB + t*AC where t and s are coefficients of the vectors AC and AB
but i don't know how that helps

 

[Mark44]

No, I mean find the equation of the plane. The cross product of AB and AC will give you a normal to the plane, say <n₁, n₂, n₃>. Then use any of the points on the plane and the normal to find the equation of the plane.

When you have the equation of the plane, determine whether your line satisfies the plane equation.