Vectors. Determine the eqn. of plane M?

[bob29]

Homework Statement

Question is:
Given the vectors a= i - j + 2k and b= 2i + 3j - k, determine the equation of the plane M that contains the vectors a and b and the point P(2,1,-1).

Homework Equations

x = x_o + at
y = y_o + bt
z = z_o + ct
P_oP = tv ---> (x-x_o)i + (y-y_o)j + (z-z_o)k = t(ai + bj + ck)

where P_o (x_o, y_o, z_o)
P(x, y, z)

The Attempt at a Solution

pt= (2,1,-1)
direction: v = ab= i + 4j - 3k

equation of line:
x=x_o + at --> x= 2 + t
y=y_o + bt --> y= 1 + 4t
z=z_o + ct --> z= -1 - 3t

and that's where I get stuck, and I don't know how to continue.

The answer is -x + y + z = -2.
I am unsure how to get to that answer.

Got the answer nevermind.

 

[HallsofIvy]

You have a lot about vectors and lines there, but nothing about planes. You need to know two fundamental facts:
1) The plane with normal vector $a\vec{i}+ b\vec{j}+ c\vec{k}$ and containing point $(x_0, y_0, z_0)$ has equation $a(x- x_0)+ b(y- y_0)+ c(z- z_0)= 0$.

2) A normal vector to the plane containing vectors $a\vec{i}+ b\vec{j}+ c\vec{k}$ and $d\vec{i}+ e\vec{j}+ f\vec{k}$ is the cross product of the two vectors $(bf- ce)\vec{i}- (cd- af)\vec{j}+ (ae- bd)\vec{j}$ which can be remembered by writing it (as a mnemonic) as if it were a determinant:
$$\left|\begin{array} {ccc}\vec{i} & \vec{j} & \vec{k} \\ a & b & c \\ d & e & f\end{array}\right|$$