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

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

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

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

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

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

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