Show that the line joining the origin to the point $(2,1,1)$ is perpendicular to the line determined by the points $(3,5,-1)$ and $(4,3,-1).$
(N/A) Let $OA$ be the line joining the origin $O(0,0,0)$ and the point $A(2,1,1).$
The direction ratios of line $OA$ are $(2-0, 1-0, 1-0),$ which are $2, 1, 1.$
Let $BC$ be the line joining the points $B(3,5,-1)$ and $C(4,3,-1).$
The direction ratios of line $BC$ are $(4-3, 3-5, -1-(-1)),$ which are $1, -2, 0.$
Two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are perpendicular if $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0.$
Substituting the values: $(2)(1) + (1)(-2) + (1)(0) = 2 - 2 + 0 = 0.$
Since the sum of the products of the direction ratios is $0,$ the line $OA$ is perpendicular to the line $BC.$
The direction ratios of line $OA$ are $(2-0, 1-0, 1-0),$ which are $2, 1, 1.$
Let $BC$ be the line joining the points $B(3,5,-1)$ and $C(4,3,-1).$
The direction ratios of line $BC$ are $(4-3, 3-5, -1-(-1)),$ which are $1, -2, 0.$
Two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are perpendicular if $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0.$
Substituting the values: $(2)(1) + (1)(-2) + (1)(0) = 2 - 2 + 0 = 0.$
Since the sum of the products of the direction ratios is $0,$ the line $OA$ is perpendicular to the line $BC.$
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