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Question

If a,b,c are three mutually perpendicular unit vectors and d is a unit vector making equal angles with a,b,c, then a+b+c+d2 is

A
4
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B
4±3
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C
4±23
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D
None of these
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Solution

The correct option is B 4±23
We have
|¯¯¯a+¯¯b+¯¯c+¯¯¯d|2=4+2[¯¯¯a.¯¯b+¯¯b.¯¯c+¯¯c.¯¯¯d.+¯¯¯d.¯¯¯a+¯¯¯a.¯¯c+¯¯b.¯¯¯d]
=4+2(0+0+0+¯¯¯d(¯¯¯a+¯¯b+¯¯c))
Let ¯¯¯d=λ¯¯¯a+μ¯¯b+v¯¯c
¯¯¯d.¯¯¯a=λ,¯¯¯d.¯¯b=μ,¯¯¯d. ¯¯c=v
But ¯¯¯d.¯¯¯a=¯¯¯d.¯¯b=¯¯¯d.¯¯c=cosθ
But λ2+μ2+v2=1
3cos2θ=1
cosθ=(±13)
|¯¯¯a+¯¯b+¯¯c+¯¯¯d|2=4+2. (±13)
=4±23

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