Points and lines: shortest distance problems

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Points and lines shortest distance problems

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Points to lines
The shortest distance from a point to a line is
an often asked examination question. The
shortest distance from a point to a line will
always be the distance of the perpendicular line
from the point to the line.
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Finding the point on the line
Calculate QP p-q
A line in space is defined as
Find the point on the line,Q, that is at the foot
of the perpendicular from the point P(3,1,-2).
The vector QP.r0, as they are perpendicular. Use
this to find .
Answer draw a quick diagram to visualise the
problem.
Substitute into the coordinates of Q.
Write the equation of the line in parametric form
and this gives the coordinate of the point Q
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Shortest distance problems
Now we can find shortest distance problems. Once
we have two points then the shortest distance
will be the magnitude between those two points.
For example the shortest distance from the point
P(3,1,-2) to the line r, defined below
In the previous slide we found the point on the
line
Find the magnitude between the two points P and
Q.