Practice section 4 of 7

A plane passes through $(1,-2,1)$ and $(2,-1,-1)$ and is parallel to the line $\mathbf r=(1,-1,0)+s(3,1,-2)$, $s\in\mathbb R$.

(i) Find the foot of the perpendicular from $A(1,1,-1)$ to the plane.

(ii) Hence or otherwise find the shortest distance from A to the plane.

(iii) Find the image of A reflected in the plane.

Given $l:\mathbf r=(10,4,2)+\lambda(9,-1,0)$, $\lambda\in\mathbb R$, and $\Pi:\mathbf r\cdot(3,-1,2)=2$, find

(i) the point of intersection of l and $\Pi$;

(ii) a vector equation of the reflection of l in $\Pi$.

For each pair of planes, find their line of intersection and acute angle. All parameters are real.

(a) $\mathbf r\cdot(1,-1,3)=3$ and $\mathbf r\cdot(2,-1,2)=5$.

(b) $\mathbf r=(1,2,7)+\lambda(1,1,0)+\mu(1,0,1)$ and $\mathbf r=(1,2,7)+s(1,2,4)+t(0,1,1)$.

The planes $p_1:\mathbf r\cdot(2,-2,1)=1$ and $p_2:\mathbf r\cdot(-6,3,2)=-1$ meet in line l.

(i) Find the acute angle between the planes.

(ii) Find a vector equation for l.

(iii) $A(4,3,c)$ is equidistant from the two planes. Calculate both possible values of c.

The planes $p_1,p_2,p_3$ are $2x-5y+3z=3$, $3x+2y-5z=-5$, $5x+\lambda y+17z=\mu$, respectively. Here $\lambda,\mu$ are constants. When $\lambda=-20.9$, $\mu=16.6$, find x,y,z satisfying all three equations.

The first two planes meet in a line l.

(i) Find a vector equation of l.

(ii) If $p_3$ contains l, find $\lambda,\mu$.

(iii) If instead $p_3$ does not intersect l, what can be said about $\lambda,\mu$?

(iv) Find a Cartesian equation of the plane containing l and $(1,-1,3)$.

(a) Points P,Q,R have position vectors $\mathbf p,\mathbf q,\mathbf r$. P and Q are fixed; R varies. Given $\mathbf r=(x,y,z)$, $\mathbf p=(-1,2,4)$, $\mathbf q=(3,-5,2)$ and $(\mathbf r-\mathbf p)\cdot\mathbf q=0$, find the relation between x,y,z and describe the locus of R.

(b) Interpret $\mathbf r\cdot\mathbf n=d$ geometrically, where $\mathbf n$ is a constant unit vector and d is a constant scalar. State what d represents.

(c) Given $\mathbf b\cdot\mathbf n\ne0$, solve $\mathbf r=\mathbf a+t\mathbf b$ and $\mathbf r\cdot\mathbf n=d$ for t in terms of $\mathbf a,\mathbf b,\mathbf n,d$. Interpret the solution geometrically.