Class 12MathematicsTHREE DIMENSIONAL GEOMETRYSystem of co-ordinates, Direction cosines and direction ratios, Projection
MediumIf $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ are the direction cosines of two mutually perpendicular lines,show that the direction cosines of the line perpendicular to both of these are $m_{1} n_{2}-m_{2} n_{1}, n_{1} l_{2}-n_{2} l_{1}, l_{1} m_{2}-l_{2} m_{1}$.
(N/A) It is given that $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ are the direction cosines of two mutually perpendicular lines. Therefore,
$l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}=0$ ........$(1)$
$l_{1}^{2}+m_{1}^{2}+n_{1}^{2}=1$ ..........$(2)$
$l_{2}^{2}+m_{2}^{2}+n_{2}^{2}=1$ ...........$(3)$
Let $l, m, n$ be the direction cosines of the line which is perpendicular to the line with direction cosines $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$.
$\therefore l l_{1} + m m_{1} + n n_{1} = 0$
$l l_{2} + m m_{2} + n n_{2} = 0$
$\therefore \frac{l}{m_{1} n_{2} - m_{2} n_{1}} = \frac{m}{n_{1} l_{2} - n_{2} l_{1}} = \frac{n}{l_{1} m_{2} - l_{2} m_{1}}$
$\Rightarrow \frac{l^{2}}{(m_{1} n_{2} - m_{2} n_{1})^{2}} = \frac{m^{2}}{(n_{1} l_{2} - n_{2} l_{1})^{2}} = \frac{n^{2}}{(l_{1} m_{2} - l_{2} m_{1})^{2}}$
$= \frac{l^{2} + m^{2} + n^{2}}{(m_{1} n_{2} - m_{2} n_{1})^{2} + (n_{1} l_{2} - n_{2} l_{1})^{2} + (l_{1} m_{2} - l_{2} m_{1})^{2}}$ .........$(4)$
Since $l, m, n$ are the direction cosines of the line,$l^{2} + m^{2} + n^{2} = 1$ ........$(5)$
Using the Lagrange identity:
$(l_{1}^{2} + m_{1}^{2} + n_{1}^{2})(l_{2}^{2} + m_{2}^{2} + n_{2}^{2}) - (l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2})^{2} = (m_{1} n_{2} - m_{2} n_{1})^{2} + (n_{1} l_{2} - n_{2} l_{1})^{2} + (l_{1} m_{2} - l_{2} m_{1})^{2}$
From $(1), (2),$ and $(3),$ we obtain:
$1 \cdot 1 - 0^{2} = (m_{1} n_{2} - m_{2} n_{1})^{2} + (n_{1} l_{2} - n_{2} l_{1})^{2} + (l_{1} m_{2} - l_{2} m_{1})^{2} = 1$ .........$(6)$
Substituting $(5)$ and $(6)$ in $(4),$ we get:
$\frac{l^{2}}{(m_{1} n_{2} - m_{2} n_{1})^{2}} = \frac{m^{2}}{(n_{1} l_{2} - n_{2} l_{1})^{2}} = \frac{n^{2}}{(l_{1} m_{2} - l_{2} m_{1})^{2}} = 1$
Thus,$l = m_{1} n_{2} - m_{2} n_{1}, m = n_{1} l_{2} - n_{2} l_{1}, n = l_{1} m_{2} - l_{2} m_{1}$.
$l_{1} l_{2}+m_{1} m_{2}+n_{1} n_{2}=0$ ........$(1)$
$l_{1}^{2}+m_{1}^{2}+n_{1}^{2}=1$ ..........$(2)$
$l_{2}^{2}+m_{2}^{2}+n_{2}^{2}=1$ ...........$(3)$
Let $l, m, n$ be the direction cosines of the line which is perpendicular to the line with direction cosines $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$.
$\therefore l l_{1} + m m_{1} + n n_{1} = 0$
$l l_{2} + m m_{2} + n n_{2} = 0$
$\therefore \frac{l}{m_{1} n_{2} - m_{2} n_{1}} = \frac{m}{n_{1} l_{2} - n_{2} l_{1}} = \frac{n}{l_{1} m_{2} - l_{2} m_{1}}$
$\Rightarrow \frac{l^{2}}{(m_{1} n_{2} - m_{2} n_{1})^{2}} = \frac{m^{2}}{(n_{1} l_{2} - n_{2} l_{1})^{2}} = \frac{n^{2}}{(l_{1} m_{2} - l_{2} m_{1})^{2}}$
$= \frac{l^{2} + m^{2} + n^{2}}{(m_{1} n_{2} - m_{2} n_{1})^{2} + (n_{1} l_{2} - n_{2} l_{1})^{2} + (l_{1} m_{2} - l_{2} m_{1})^{2}}$ .........$(4)$
Since $l, m, n$ are the direction cosines of the line,$l^{2} + m^{2} + n^{2} = 1$ ........$(5)$
Using the Lagrange identity:
$(l_{1}^{2} + m_{1}^{2} + n_{1}^{2})(l_{2}^{2} + m_{2}^{2} + n_{2}^{2}) - (l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2})^{2} = (m_{1} n_{2} - m_{2} n_{1})^{2} + (n_{1} l_{2} - n_{2} l_{1})^{2} + (l_{1} m_{2} - l_{2} m_{1})^{2}$
From $(1), (2),$ and $(3),$ we obtain:
$1 \cdot 1 - 0^{2} = (m_{1} n_{2} - m_{2} n_{1})^{2} + (n_{1} l_{2} - n_{2} l_{1})^{2} + (l_{1} m_{2} - l_{2} m_{1})^{2} = 1$ .........$(6)$
Substituting $(5)$ and $(6)$ in $(4),$ we get:
$\frac{l^{2}}{(m_{1} n_{2} - m_{2} n_{1})^{2}} = \frac{m^{2}}{(n_{1} l_{2} - n_{2} l_{1})^{2}} = \frac{n^{2}}{(l_{1} m_{2} - l_{2} m_{1})^{2}} = 1$
Thus,$l = m_{1} n_{2} - m_{2} n_{1}, m = n_{1} l_{2} - n_{2} l_{1}, n = l_{1} m_{2} - l_{2} m_{1}$.
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