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

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

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

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.

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 .

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