Find the vector equation of the plane that contains the points A(2,5,6) B(-7,1,4) and C(6,-2,-9).?

Does it contain the line with equation


[x,y,z] = [-3,-6,-11]+k[22,1,-11]? varify.

Find the vector equation of the plane that contains the points A(2,5,6) B(-7,1,4) and C(6,-2,-9).?
Given points A(2,5,6); B(-7,1,4); and C(6,-2,-9), find the vector equation of the plane containing them.





Let's create two vectors, u and v, from the points.





u = AB = %26lt;-7-2, 1-5, 4-6%26gt; = %26lt;-9, -4, -2%26gt;


v = AC = %26lt;6-2, -2-5, -9-6%26gt; = %26lt;4, -7, -15%26gt;





The vector equation of the plane P is:





P = A + su + tv = %26lt;2, 5, 6%26gt; + s%26lt;-9, -4, -2%26gt; + t%26lt;4, -7, -15%26gt;


P = %26lt;2 - 9s + 4t, 5 - 4s - 7t, 6 - 2s - 15t%26gt;


where s and t are scalars ranging over the real numbers.


___________________________





Does the plane P contain the line L with the equation





r = %26lt;-3,-6,-11%26gt; + k%26lt;22,1,-11%26gt;


where k is a scalar ranging over the real numbers?





If the given point Q(-3, -6, -11) is in the plane and the vector %26lt;22,1,-11%26gt; lies in the plane, then the line is in the plane.





First let's check the point.





P = %26lt;2 - 9s + 4t, 5 - 4s - 7t, 6 - 2s - 15t%26gt;


Q(-3, -6, -11)





Set each variable of the plane equal to the corresponding value for the point.





x: 2 - 9s + 4t = -3


y: 5 - 4s - 7t = -6


z: 6 - 2s - 15t = -11





y - 2z: -7 + 23t = 16


23t = 23


t = 1





Plug back into z to solve for s.


z: 6 - 2s - 15t = -11


6 - 2s - 15*1 = -11


-2s = -11 - 6 + 15 = -2


s = 1





Now plug into x to check for consistency.


x: 2 - 9s + 4t = -3


x: 2 - 9 + 4 = -3


-3 = -3





This is true so the point Q(-3, -6, -11) lies in the plane.





Now let's check the vector.





P = %26lt;2 - 9s + 4t, 5 - 4s - 7t, 6 - 2s - 15t%26gt;


Q(-3, -6, -11)





We have three equations in three unknowns. Notice we are only setting the components of the vectors equal.





x: - 9s + 4t = 22k


y: - 4s - 7t = 1k


z: - 2s - 15t = -11k





y - 2z: 23t = 23k


t = k





Plug back into z to solve for s.





z: - 2s - 15t = -11k


z: - 2s - 15k = -11k


-2s = 4k


s = -2k





Plug into x for consistency check.





x: - 9s + 4t = 22k


x: - 9(-2k) + 4k = 22k


22k = 22k





The vector also lies in the plane.





The point and vector both lie in the plane so the line lies in the plane.