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Definition of Vector

The direction...

Question

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

A

1

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B

0

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C

4

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D

2

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Solution

The correct option is A $1$
We have $(l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2})^{2} + \sum (m_{1} n_{2} - m_{2} n_{1})^{2}$
$=$ $(l_{1} l_{2})^{2} + (m_{1} m_{2})^{2} + (n_{1} n_{2})^{2} + 2 \times (l_{1} l_{2} m_{1} m_{2}) + 2 \times (m_{1} m_{2} n_{1} n_{2}) + 2 \times (l_{1} l_{2} n_{1} n_{2}) + (m_{1} n_{2})^{2}$
$+ (m_{2} n_{1})^{2} - 2 \times (m_{1} m_{2} n_{1} n_{2}) + (l_{1} n_{2})^{2} + (l_{2} n_{1})^{2} - 2 \times (l_{1} l_{2} n_{1} n_{2}) + (l_{1} m_{2})^{2}$
$+ (l_{2} m_{1})^{2} - 2 \times (l_{1} l_{2} m_{1} m_{2})$

$=$ $(l_{1} l_{2})^{2} + (m_{1} m_{2})^{2} + (n_{1} n_{2})^{2} + (m_{1} n_{2})^{2} + (m_{2} n_{1})^{2} + (l_{1} n_{2})^{2} + (l_{2} n_{1})^{2}$
$+ (l_{1} m_{2})^{2} + (l_{2} m_{1})^{2}$

$=$ $l_{1}^{2} (l_{2}^{2} + m_{2}^{2} + n_{2}^{2}) + m_{1}^{2} (l_{2}^{2} + m_{2}^{2} + n_{2}^{2}) + n_{1}^{2} (l_{2}^{2} + m_{2}^{2} + n_{2}^{2})$
$=$ $l_{1}^{2} + m_{1}^{2} + n_{1}^{2}$
$=$ $1$ (Direction cosines' square sums up to 1)

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Similar questions

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

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} =

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

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 -

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