If a ray makes angles a, b, y, 8 with the four diagonals of a cube find
cos a + cosB + cos” y + cos? 8
Answers
Step-by-step explanation:
REF.Image
Take 'O' as a corner
OA,OB,OC are 3 edges through the axes
Let OA=OB=OC=a
coordinates of O=(o,o,o)
A(a,o,o)B(o,a,o)C(o,o,a)
P(a,a,o)L(o,a,a)M(a,o,a)N(a,a,o)
The four diagonals OP,AL,BM,CN
Direction cosine of OP:a−o,a−o,a−o=a,a,a=1,1,1
Direction cosine of AL:o−a,a−o,a−o=−a,a,a=−1,1,1
Direction cosine of BM:a−o,o−a,a−o=a,−a,a=1,−1,1
Direction cosine of CN:a−o,a−o,o−a=a,a,−a=1,1,−1
∴ DC's of OP are
3
1
,
3
1
,
3
1
DC's of AL are
3
−1
,
3
1
,
3
1
DC's of BM are
3
1
,
3
−1
,
3
1
DC's of CN are
3
1
,
3
1
,
3
−1
Let l,m,n be dc's of line and line makes angle α
with OP :- cosα=l(
3
1
)+m(
3
1
)+n(
3
1
)=
3
l+m+n
Similarly cosβ=
3
−l+m+n
cosδ=
3
l+m−n
cosγ=
3
l−m+n
suaring and adding all the four
i.e ; cos
2
α+cos
2
β+cos
2
γ+cos
2
δ
=
3
1
[(l+m+n)
2
+(−l+m+n)
2
+(l−m+n)
2
+(l+m−n)
2
]
=
3
1
[4l
2
+4m
2
+4n
2
]=
3
4
(l
2
+m
2
+n
2
)
[∵l
2
+m
2
+n
2
=1]=
3
4
∴cos
2
α+cos
2
β+cos
2
γ+cos
2
δ=
3
4