Vector distance formula

by linuspauling
Tags: distance, formula, vector
 P: 11 let P be a point NOT on line L that passes through points Q and R. $$\vec{A} = QR$$ $$\vec{B} = QP$$ prove that distance from point P to anywhere on line L is $$d = |\vec{A} x \vec{B}| divided by |\vec{A}|$$ so, i've tried doing the cross product after assigning variables for the A and B components. I ended up with a very tedious long multiplication of several variables, and I was wondering if there is an easier way to prove this formula.
 P: 406 Draw a picture of what is going on and note that |AxB| is the area of the parallelogram generated by A and B. It's also equal to |A||B|Sin(t) where t is the angle between A and B.
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,898 Of course the shortest distance from P to a line is along the line through P perpendicular to the line. You might start by finding the equation of a line through P perpendicular to $\vec{QR}$.

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