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Dot Product of Vectors

The dot product (also known as the scalar product) is an operation on vectors that can tell you the angle between the vectors.

Dot Product - Distance between Point and a Line

         

Find the shortest distance between the point \(P=(1,-2,3)\) and the line \[2(x-3)=y-1=z+1.\]

Find the shortest distance between the point \(P=(0,3,4)\) and the line \[x=0,\ \frac{y}{2}=-z+3.\]

Find the shortest distance between the origin and the line \[x-1=\frac{y+1}{-4}=\frac{z-7}{3}.\]

What is the shortest distance between the line \(x-2=y-1=z+1\) and the sphere \(x^2+y^2+z^2=4?\)

Find the shortest distance between the origin and the line \[\frac{x+1}{2}=-y+2=-z+2.\]

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