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
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.
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Shortest distance from a point to a plane
Find the shortest distance from the point
P(1,2,3) to the plane 3x2y-4z53.
To solve these questions 1. Find the planes
normal vector, n. 2. Find the parametric
equations of the normal vector. 3. Substitute the
parametric equations into the equation of the
plane, and find . 4. Use this value to find the
exact coordinates of on the plane where the
normal from P meets the plane.
Substitute the parametric equations into the
equation of the plane, and find .
Find the coordinates of N.
(7,6,-5)
Answer
Find the magnitude of NP.
The normal vector to the plane is defined as
3i2j-4k.
Let N be the point on the plane where the normal
from P meets the plane. NP has the vector of the
normal and the parametric equations of NP are