Question

Ques Find the direction ratios of a line perpendicular to the two lines determined by the pairs of points
(2.3.-4).(-3,3,-2) and (-1,4,2), (3,5,1).
​

Answer 1

The direction ratios of the line passing through the points (2, 3, -4) and (-3, 3, -2) are,

$\longrightarrow\vec{b_1}=\lambda_1\left<2-(-3),\ 3-3,\ (-4)-(-2)\right>$

$\longrightarrow\vec{b_1}=\lambda_1\left<5,\ 0,\ -2\right>$

The direction ratios of the line passing through the points (-1, 4, 2) and (3, 5, 1) are,

$\longrightarrow\vec{b_2}=\lambda_2\left<3-(-1),\ 5-4,\ 1-2\right>$

$\longrightarrow\vec{b_2}=\lambda_2\left<4,\ 1,\ -1\right>$

The direction ratios of the line perpendicular to these two lines are parallel to the cross product of $\vec{b_1}$ and $\vec{b_2},$ i.e.,

$\longrightarrow\vec{b}=\mu\left(\vec{b_1}\times\vec{b_2}\right)$

$\longrightarrow\vec{b}=\mu\lambda_1\lambda_2\left|\begin{array}{ccc}\hat i&\hat j&\hat k\\5&0&-2\\4&1&-1\end{array}\right|$

$\longrightarrow\vec{b}=\mu\lambda_1\lambda_2\left<2,\ -3,\ 5\right>$

Taking $\mu\lambda_1\lambda_2=\lambda,$ we get the direction ratios of the required line.

$\longrightarrow\underline{\underline{\vec{b}=\lambda\left<2,\ -3,\ 5\right>}}$

for some non - zero real numbers $\mu,\ \lambda_1$ and $\lambda_2.$