find the direction cosines of the sides of the triangle whose vertices are ( 3,5,-4),(-1,1,2)and( -5,-5,-2)
Answers
Answer:
Answer
Let, A(3,5,−4),B(−1,1,2) and C(−5,−5,−2)
(i) Direction ratios of the line joining A and B are
(−1−3),(1−5),(2−(−4))=(−4,−4,6)
∴AB=
(−4)
2
+(−4)
2
+6
2
=
16+16+36
=
68
=2
17
The direction cosines are
2
17
−4
,
2
17
−4
,
2
17
6
=
17
−2
,
17
−2
,
17
3
(ii) Direction ratios of the line joining B and C are
(−5−(−1)),(−5−1),(−2−2)=(−4,−6,−4)
∴BC=
(−4)
2
+(−6)
2
+(−4)
2
=
16+36+16
=
68
=2
17
The direction cosines are
2
17
−4
,
2
17
−6
,
2
17
−4
=
17
−2
,
17
−3
,
17
−2
(iii) Direction ratios of the line joining C and A are
(3−(−5)),(5−(−5)),(−4−(−2))=(8,10,−2)
∴CA=
(8)
2
+(10)
2
+(−2)
2
=
64+100+4
=
168
=2
42
The direction cosines are
2
42
8
,
2
42
10
,
2
42
−2
=
42
4
,
42
5
,
42
−1
Two of the sides are equal, so the triangle is an isosceles triangle.