find the projection of point p=(3,-1,4) on l=(2,0,3)+t(1,5,2)
x∈l:x=(2+t,5t,3+2t)∥p−x∥2→min(3−2−t)2+(−1−5t)2+(4−3−2t)2→min30t2+4t+3→mint=−42⋅30=−115x=(2−115,−5⋅115,3−2⋅115)=(2915,−13,4315)x\in l:x=\left( 2+t,5t,3+2t \right) \\\left\| p-x \right\| ^2\rightarrow \min \\\left( 3-2-t \right) ^2+\left( -1-5t \right) ^2+\left( 4-3-2t \right) ^2\rightarrow \min \\30t^2+4t+3\rightarrow \min \\t=-\frac{4}{2\cdot 30}=-\frac{1}{15}\\x=\left( 2-\frac{1}{15},-5\cdot \frac{1}{15},3-2\cdot \frac{1}{15} \right) =\left( \frac{29}{15},-\frac{1}{3},\frac{43}{15} \right) \\x∈l:x=(2+t,5t,3+2t)∥p−x∥2→min(3−2−t)2+(−1−5t)2+(4−3−2t)2→min30t2+4t+3→mint=−2⋅304=−151x=(2−151,−5⋅151,3−2⋅151)=(1529,−31,1543)
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