Question #314906

find the projection of point p=(3,-1,4) on l=(2,0,3)+t(1,5,2)


1
Expert's answer
2022-03-21T00:37:15-0400

xl:x=(2+t,5t,3+2t)px2min(32t)2+(15t)2+(432t)2min30t2+4t+3mint=4230=115x=(2115,5115,32115)=(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) \\


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