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Work Done as Dot Product

l1,m1,n1 and ...

Question

$(l_{1}, m_{1}, n_{1})$ and $(l_{2}, m_{2}, n_{2})$ are d.c's two lines. Then
The lines are parallel is $\frac{l_{1}}{l_{2}} = \frac{m_{1}}{m_{2}} = \frac{n_{1}}{n_{2}}$

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Solution

Let $L_{1}$ has d.cs $l_{1}, m_{1}, n_{1} \Rightarrow l_{1}$ is parallel to $l_{1}^i + m_{1}^j + n_{1}^k$
Let $L_{2}$ has d.cs $l_{2}, m_{2}, n_{2} \Rightarrow l_{2}$ is parallel to $l_{2}^i + m_{2}^j + n_{2}^k$
If $_{1} | |_{2} \Rightarrow_{2} = λ_{1} \Rightarrow l_{2}^i + m_{2}^j + n_{2}^k = λ (l_{1}^i + m_{1}^j + n_{1}^k)$
$\Rightarrow l_{2} = l_{1} λ, m_{2} = λ m_{1}, n_{2} = λ n_{1}$
$\Rightarrow \frac{l_{1}}{l_{2}} = \frac{m_{1}}{m_{2}} = \frac{n_{1}}{n_{2}} = \frac{1}{λ}$ .

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Similar questions

(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

If two lines have direction cosines

$l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$

then the condition for these lines being perpendicular to each other would be -

Q. If two lines have direction cosines

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

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

then the condition for these lines being parallel to each other would be -

Q. If two lines with direction cosines

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

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

intersect, then the angle between them would be -

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}

.

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