Practice section 2 of 7

Find the acute angle between

and the plane $\mathbf r\cdot(2\mathbf i-\mathbf j+\mathbf k)=4$, to the nearest $0.1^\circ$.

A plane $p$ has equation $3x-3y+2z=3$. Show that $Q(1,2,3)$ lies on $p$.

Find the foot of the perpendicular N from $A(1,3,-1)$ to the plane $p:\ \mathbf r\cdot(0,-1,2)=5$.

Find a vector equation of the line through $A(3,1,2)$ perpendicular to the plane $\mathbf r\cdot(2\mathbf i-\mathbf j+\mathbf k)=4$.

(i) Find the intersection of the line and plane.

(ii) State the shortest distance from the line to the plane.

(iii) Find the shortest distance from A to the plane.

Two planes have equations

Explain why they are parallel and find the distance between them.

Find the equation of the line of intersection of $\mathbf r\cdot(\mathbf i+\mathbf j-3\mathbf k)=6$ and $\mathbf r\cdot(2\mathbf i-\mathbf j+\mathbf k)=4$.