How close does the plane \(2x+y-2z=9\) come to the origin?
What’s the distance from the point \(\bigl(0,11,-1\bigr)\)
to the plane \({2x-9y+6z=16?}\)
Which of the points \((-16,8,12)\) or \((7,11,-10)\) or \((10,10,10)\)
is closest to the plane \(-3x+7y+z=19?\)
How close does the line
\(\bigl(-1,6,-1\bigr) + \bigl\langle 2,-4,3 \bigr\rangle t \)
come to the origin?
What’s the shortest distance from the point \(\bigl(4,-1,-6\bigr)\)
to the line \({\bigl(0,3,13\bigr) + \bigl\langle 1,0,-2 \bigr\rangle t\,?}\)
Which of the points \((6,-7,8)\) or \((1,-8,2)\) or \((-2,-7,-1)\)
is closest to the line \(\bigl(9,-4,9\bigr) + {\bigl\langle 3,1,2 \bigr\rangle t ?}\)
What’s the distance between the line
\({\bigl(1,5,2\bigr) + \bigl\langle -1,4,1 \bigr\rangle t }\)
and the \(z\)-axis?
What’s the shortest distance between the lines
\({\bigl(-6,5,-6\bigr) + \bigl\langle 4,2,3 \bigr\rangle t }\)
and \({\bigl(5,-3,4\bigr) + \bigl\langle 4,2,3 \bigr\rangle t\,?}\)
What’s the shortest distance between the lines
\({\bigl(-2,8,0\bigr) + \bigl\langle 0,1,3 \bigr\rangle t}\)
and \({\bigl(-3,-6,1\bigr) + \bigl\langle -4,-1,0 \bigr\rangle t\,?}\)
Problems & Challenges
Devise a formula for the distance
from a point \(P\) to a plane \({Ax\!+\!By\!+\!Cz\!=\!D.}\)
Devise a formula for the distance
from a point \(P\) to a line \({Q\!+\!\bm{v}t.}\)
Devise a formula for the distance
between the lines \({P\!+\!\bm{u}t}\) and \({Q\!+\!\bm{v}t.}\)