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Question

If a line makes α,β,γ and δ from the diagonals of a cube then cos2α+cos2β+cos2γ+cos2δ=a3. Find a

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Solution


LetOA=OB=OC=a
Then the co-ordinates of O,A,B&C are
(0,0,0),(a,0,0),(0,a,0),(0,0,a). and that of P,L,MandN are(a,a,a),(0,a,a),(0,0,a),(a,a,0) respectively.
The four diagonals are UP,AL,BM,CN.
Direction cosines of OP are proportional toa0,a0,a0i.e,(a,0,a),i.e.,1,1,1.
Similarily,
Direction cosines of AL are proportional to 0a,a0,a0i.e.,1,1,1.
Direction cosines of BM are proportional to 1,1,1
Direction cosines of OP are 13,13,13
Direction cosines of AL are 13,13,13
Direction cosines of BM are 13,13,13
Direction cosines of CN are 13,13,13
Let l,m,n be direction cosine of the line.
The line makes angle α with OP.
cosα=l(13)+m(13)+n(13)
cosα=l+m+n3(i)
Similarily,cosβ=l+m+n3(ii)
andcosγ=lm+n3(iii)
cosδ=l+mn3(iv)
Squaring and adding (i),(ii),(iii)&(iv)
cos2α+cos2β+cos2γ+cos2δ
=13[(l+m+n)2+(l+m+n)2+(lm+n)2+(l+mn)2]
=13[4l2+4m2+4n2]=43[l2+m2+n2]
=43(1)=43(l2+m2+n2=1)
a=4

1024136_1035798_ans_672746f1fd324258a66d511ca2bc9b44.png

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