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Question: Find the equation of lines passing through the point \[\left( 3,1,2 \right)\]and perpendicular to th...

Find the equation of lines passing through the point (3,1,2)\left( 3,1,2 \right)and perpendicular to the lines x−11=y−22=z−33\dfrac{x-1}{1}=\dfrac{y-2}{2}=\dfrac{z-3}{3}and x−3=y2=z5\dfrac{x}{-3}=\dfrac{y}{2}=\dfrac{z}{5}

Explanation

Solution

Hint: we know that when two lines are perpendicular to each other then the condition for their direction ratios is a1a2+b1b2+c1c2=0{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}=0. Now wee will get two equations and by cross multiplication we will get the direction ratios of the required line then we know the equation of line passing through a point and having direction ratios a, b, c.

Complete step-by-step answer:
Let a, b, c be the direction ratios of required ratios of the required line which is perpendicular to the lines x−11=y−22=z−33\dfrac{x-1}{1}=\dfrac{y-2}{2}=\dfrac{z-3}{3}and x−3=y2=z5\dfrac{x}{-3}=\dfrac{y}{2}=\dfrac{z}{5}
We know that when two lines are perpendicular to each other then the condition for direction cosines is a1a2+b1b2+c1c2=0{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}=0
a+2b+3c=0a+2b+3c=0. . . . . . . . . . . . . . . . . . . (1)
−3a+2b+5c=0-3a+2b+5c=0.. . . . . . . . . . . . . . . . (2)
Do the cross multiplication then we will get

2 & 3 \\\ 2 & 5 \\\ \end{matrix} \right)}=\dfrac{-b}{\left( \begin{matrix} 1 & 3 \\\ -3 & 2 \\\ \end{matrix} \right)}=\dfrac{c}{\left( \begin{matrix} 1 & 2 \\\ -3 & 2 \\\ \end{matrix} \right)}$$ $$=\dfrac{a}{4}=\dfrac{b}{-14}=\dfrac{c}{8}$$. . . . . . . . . . . . . . . . . . (3) The direction ratios of required line is 4, -14, 8 or 2, -7, 4 The equation of a line passing through the point $$\left( {{x}_{1,}}{{y}_{1,}}{{z}_{1}} \right)$$and having direction ratios a, b, c is $$\dfrac{x-{{x}_{1}}}{a}=\dfrac{y-{{y}_{1}}}{b}=\dfrac{z-{{z}_{1}}}{c}$$ The equation of line passing through A $$\left( 3,1,2 \right)$$and having direction ratios 2, -7, 4 is $$\dfrac{x-3}{2}=\dfrac{y-1}{-7}=\dfrac{z-2}{4}$$ Note: The equation of line passing through a point A $$\left( {{x}_{1,}}{{y}_{1,}}{{z}_{1}} \right)$$and having direction ratios a, b, c in cartesian form is $$\dfrac{x-{{x}_{1}}}{a}=\dfrac{y-{{y}_{1}}}{b}=\dfrac{z-{{z}_{1}}}{c}$$. If two lines are parallel to each other then the direction ratios of those two lines are equal.