Q.Find the vector and Cartesian equations of the plane passing through the points with position vectors $3\vec i + 4\vec j + 2\vec k$, $2\vec i - 2\vec j - \vec k$ and $7\vec i + \vec k$.

Find two vectors lying in the plane from the three given points, take their cross product to get the normal vector, then write the plane equation through any one of the points.

1. Let $A=3\vec i+4\vec j+2\vec k$, $B=2\vec i-2\vec j-\vec k$, $C=7\vec i+0\vec j+\vec k$ be the given position vectors.
2. Vectors in the plane: $\vec{AB}=B-A=(2-3)\vec i+(-2-4)\vec j+(-1-2)\vec k = -\vec i-6\vec j-3\vec k$ $\vec{AC}=C-A=(7-3)\vec i+(0-4)\vec j+(1-2)\vec k = 4\vec i-4\vec j-\vec k$
3. Normal vector $\vec n = \vec{AB}\times\vec{AC} = \begin{vmatrix}\vec i&\vec j&\vec k\\-1&-6&-3\\4&-4&-1\end{vmatrix}$
4. $\vec n_i = (-6)(-1)-(-3)(-4) = 6-12=-6$ $\vec n_j = -[(-1)(-1)-(-3)(4)] = -[1+12]=-13$ $\vec n_k = (-1)(-4)-(-6)(4) = 4+24=28$ So $\vec{AB}\times\vec{AC} = -6\vec i-13\vec j+28\vec k$; taking the equivalent opposite normal $\vec n=6\vec i+13\vec j-28\vec k$. …