Practice section 1 of 7

Find the vector equation of each plane in parametric form. All parameters in the given lines are real.

(i) The plane contains both lines

(ii) The plane contains the point $(1,2,4)$ and the line

(iii) The plane contains $A(3,1,2)$, $B(2,1,2)$ and $C(1,2,1)$.

Find the vector equation, in scalar product form, of the plane containing the point with position vector $3\mathbf i+\mathbf j+2\mathbf k$ and normal to $\mathbf i+2\mathbf j+\mathbf k$.

For each plane, convert the parametric equation into scalar-product form and find a Cartesian equation. All parameters $s,t$ are real.

(i)

(ii)

(iii)

Find the point of intersection of the line

and the plane $\mathbf r\cdot(2,-1,2)=4$.

Show that the line

does not intersect the plane $\mathbf r\cdot(1,2,-1)=4$.

Show that the line

lies on the plane $\mathbf r\cdot(1,2,-1)=3$.