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Question

if a,b,c are mutually perpendicular vectors of equal magnitudes, show that the vector is equally inclined to a,b,c

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Solution

¯a.¯b=¯b.¯c=¯c.¯a=0
Let ¯a+¯b+¯c=¯d and ¯d makes angle α,β,γ with ¯a,¯b,¯c
cosα=¯v,¯a|v||¯a|,cosβ=¯v.¯b|v||¯b|,cosγ=¯v.¯c|v||¯c|
cosα=(¯a+¯b+¯c).¯a|v||¯a|=¯a.¯a+¯a.¯b+¯a.¯c|v||¯a|=|a|2|v||¯a|=|a||v|
cosβ=|b||v|andcosγ=|c||v|
given that |¯a|=|¯b|=|¯c|
hence cosα=cosβ=cosγ
α=β=γ
¯a+¯b+¯c is equal inclined to ¯a,¯b and ¯c

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