Answer to Question #314906 in Math for math student

Question

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

Answer 1

$x\in l:x=\left( 2+t,5t,3+2t \right) \\\left\| p-x \right\| ^2\rightarrow \min \\\left( 3-2-t \right) ^2+\left( -1-5t \right) ^2+\left( 4-3-2t \right) ^2\rightarrow \min \\30t^2+4t+3\rightarrow \min \\t=-\frac{4}{2\cdot 30}=-\frac{1}{15}\\x=\left( 2-\frac{1}{15},-5\cdot \frac{1}{15},3-2\cdot \frac{1}{15} \right) =\left( \frac{29}{15},-\frac{1}{3},\frac{43}{15} \right) \\$