If $l_{1}, m_{1}, n_{1} a n d l_{2}, m_{2}, n_{2}$ are the direction cosines of two mutually perpendicular lines, show that the direction consines of the line perpendicular to both of these are $m_{1} n_{2} - m_{2} n_{1}, n_{1} l_{2} - n_{2} l_{1}, l_{1} m_{2} - l_{2} m_{1}$

Open in App

Solution

Given, lines are respectively parallel to unit vector
$b_{1} = l_{1}^i + m_{1}^j + n_{1}^k$ . . .(i)
and $b_{2} = l_{2}^i + m_{2}^j + n_{2}^k$ . . .(ii)
NOW $b_{1} \times b_{2}$ is a vector which is at right angles to both $b_{1} a n d b_{2}$ and is of magnitude unity.
Hence, components of $b_{1} \times b_{2}$ are direction cosines of a line which is at right angles to both $b_{1} a n d b_{2}$ . So, we compute $b_{1} \times b_{2}$
$b_{1} \times b_{2} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = (m_{1} n_{2} - m_{2} n_{1})^i - (n_{2} l_{1} - n_{1} l_{2})^j + (l_{1} m_{2} - l_{2} m_{1})^k = (m_{1} n_{2} - m_{2} n_{1})^i + (n_{1} l_{2} - n_{2} l_{1})^j + (l_{1} m_{2} - l_{2} m_{1})^k$
Thus, the direction consines of the required line are
$m_{1} n_{2} - m_{2} n_{1}, n_{1} l_{2} - n_{2} l_{1}, l_{1} m_{2} - l_{2} m_{1}$

Suggest Corrections

Similar questions

Q. If

l_{1}, m_{1}, n_{1}

and

l_{2}, m_{2}, n_{2}

are the direction cosines of two lines, then show that the direction cosines of the line perpendicular to them are proportional to

m_{1} n_{2} - m_{2} n_{1}

n_{1} l_{2} - n_{2} l_{1}

l_{1} m_{2} - l_{2} m_{1}

Q. If

l_{1}, m_{1}, n_{1}

and

l_{2}, m_{2}, n_{2}

are the direction cosines of two perpendicular lines, then the direction cosine of the line which is perpendicular to both the lines, will be

Q. If 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 .

Q. If

l_{1}, m_{1}, n_{1}

and

l_{2}, m_{2}, n_{2}

are the direction cosines of two perpendicular lines, then the direction cosine of the line which is perpendicular to both the lines, will be

If $l_{1}, m_{1}, n_{1}, l_{2}, m_{2}, n_{2} a n d l_{3}, m_{3}, n_{3}$ are the direction cosines of three mutually perpendicular lines,then prove that the line whose direction cosines are proportional to $l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3} a n d n_{1} + n_{2} + n_{3}$ makes equal angles with them.

Join BYJU'S Learning Program

If l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines, show that the direction consines of the line perpendicular to both of these are m1n2−m2n1, n1l2−n2l1, l1m2−l2m1

If $l_{1}, m_{1}, n_{1} a n d l_{2}, m_{2}, n_{2}$ are the direction cosines of two mutually perpendicular lines, show that the direction consines of the line perpendicular to both of these are $m_{1} n_{2} - m_{2} n_{1}, n_{1} l_{2} - n_{2} l_{1}, l_{1} m_{2} - l_{2} m_{1}$