Shortest distance from a line to a point

1. Finding the Shortest Distance Between a Line and a Point
Very often we will be asked to find the shortest distance between a point and a
line.
Answer the following questions about this numerical example so you can make
some general guidelines for these types of problems.
Find the shortest distance between the point P (-1,1,2) and 𝑙 = (
1
0
2
) + 𝜆 (
−3
1
1
)
𝑙
Draw a line from the point P to 𝑙
and call the point if intersection Q
Using the equation of the line what is the general expression of a point on this
line:
Now find an expression for 𝑃𝑄⃗⃗⃗⃗⃗
Explain why
𝑃𝑄⃗⃗⃗⃗⃗ ∙ (
−3
1
1
) = 0
Now use this to find a value for 𝜆
How do we find the magnitude of 𝑃𝑄⃗⃗⃗⃗⃗ ?
Explain why this is the shortest distance between point P and the line 𝑙