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Integration by Parts

If l1,m1,n1...

Question

If $l_{1}, m_{1}, n_{1}$ and $1_{2}, m_{2}, n_{2}$ are the direction cosines of two lines, then $(l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2})^{2} + \sum (m_{1} n_{2} - m_{2} n_{1})^{2} =$

A

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B

- 1

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C

1

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D

2

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Solution

The correct option is C $1$
Dot product of dc's of two lines gives,
$(l_{1}^i + m_{1}^j + n_{1}^k) . (l_{2}^i + m_{2}^j + n_{2}^k) = \sqrt{{l_{1}}^{2} + {m_{1}}^{2} + {n_{1}}^{2}} \times \sqrt{{l_{2}}^{2} + {m_{2}}^{2} + {n_{2}}^{2}} \times cos θ$
We know ${l_{1}}^{2} + {m_{1}}^{2} + {n_{1}}^{2} = 1, {l_{2}}^{2} + {m_{2}}^{2} + {n_{2}}^{2} = 1$
$\Rightarrow l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 1.1. cos θ \Rightarrow {(l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2})}^{2} = {cos}^{2} θ$
Cross product of dc's of two lines
$∣ ∣ (l_{1}^i + m_{1}^j + n_{1}^k) \times (l_{2}^i + m_{2}^j + n_{2}^k) ∣ ∣ = ∣ ∣ \sqrt{{l_{1}}^{2} + {m_{1}}^{2} + {n_{1}}^{2}} \times \sqrt{{l_{2}}^{2} + {m_{2}}^{2} + {n_{2}}^{2}} \times sin θ ∣ ∣$
$\Rightarrow \sqrt{Σ (m_{1} n_{2} - m_{2} n_{1})^{2}} = sin θ \Rightarrow Σ (m_{1} n_{2} - m_{2} n_{1})^{2} = {sin}^{2} θ$
${(l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2})}^{2} + Σ (m_{1} n_{2} - m_{2} n_{1})^{2} = {cos}^{2} θ + {sin}^{2} θ = 1$

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Similar questions

Q. The direction cosines of two lines are

(l_{1}, m_{1}, n_{1})

(l_{2}, m_{2}, n_{2})

, then the value of

(l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2})^{2} + \sum (m_{1} n_{2} - m_{2} n_{1})^{2}

Q. Assertion :The direction cosines of one of the angular bisectors of two intersecting lines having direction cosines as

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

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

are proportional to

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

. Reason: The angle between the two intersecting lines having direction cosines as

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

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

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

.

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

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

(l_{1}, m_{1}, n_{1})

(l_{2}, m_{2}, n_{2})

are d.c's two lines. Then
The lines are perpendicular if

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

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

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}

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