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The Effect of Path Difference

If ℓ 1,m1,n...

Question

If $ℓ_{1}, m_{1}, n_{1} a n d ℓ_{2}, m_{2}, n_{2}$ be direction cosines of two perpendicular lines, then

A

ℓ_{1} ℓ_{2} + m_{1} m_{2} + n_{1} n_{2} = 0

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B

ℓ_{1} ℓ_{2} + m_{1} m_{2} + n_{1} n_{2} = 1

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C

\frac{ℓ_{1}}{ℓ_{2}} + \frac{m_{1}}{m_{2}} + \frac{n_{1}}{n_{2}} = 1

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D

\frac{ℓ_{1}}{ℓ_{2}} + \frac{m_{1}}{m_{2}} + \frac{n_{1}}{n_{2}} = 0

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Solution

The correct option is A $ℓ_{1} ℓ_{2} + m_{1} m_{2} + n_{1} n_{2} = 0$

We know that ,

$cos θ = \frac{l_{1} \times l_{2} + m_{1} \times m_{2} + n_{1} \times n_{2}}{\sqrt{l_{1} + {m_{1}}^{2} + {n_{1}}^{2}} \sqrt{l_{2} + {m_{2}}^{2} + {n_{2}}^{2}}}$

If $θ = \frac{π}{2}$

Then

$cos \frac{π}{2} = \frac{l_{1} \times l_{2} + m_{1} \times m_{2} + n_{1} \times n_{2}}{\sqrt{l_{1} + {m_{1}}^{2} + {n_{1}}^{2}} \sqrt{l_{2} + {m_{2}}^{2} + {n_{2}}^{2}}}$

$0 = \frac{l_{1} \times l_{2} + m_{1} \times m_{2} + n_{1} \times n_{2}}{\sqrt{l_{1} + {m_{1}}^{2} + {n_{1}}^{2}} \sqrt{l_{2} + {m_{2}}^{2} + {n_{2}}^{2}}}$

$\frac{l_{1} \times l_{2} + m_{1} \times m_{2} + n_{1} \times n_{2}}{\sqrt{l_{1} + {m_{1}}^{2} + {n_{1}}^{2}} \sqrt{l_{2} + {m_{2}}^{2} + {n_{2}}^{2}}} = 0$

$l_{1} \times l_{2} + m_{1} \times m_{2} + n_{1} \times n_{2} = 0 \times \sqrt{l_{1} + {m_{1}}^{2} + {n_{1}}^{2}} \sqrt{l_{2} + {m_{2}}^{2} + {n_{2}}^{2}} = 0$

$l_{1} \times l_{2} + m_{1} \times m_{2} + n_{1} \times n_{2} = 0$

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