Question -
Answer -
Given:
The vertices are (3,5, –4), (-1, 1, 2) and (–5, –5, –2).
The direction cosinesof the two points passing through A(x1, y1, z1)and B(x2, y2, z2) is given by (x2 –x1), (y2-y1), (z2-z1)
Firstly let us findthe direction ratios of AB
Where, A = (3, 5, -4)and B = (-1, 1, 2)
Ratio of AB = [(x2 –x1)2, (y2 – y1)2,(z2 – z1)2]
= (-1-3), (1-5),(2-(-4)) = -4, -4, 6
Then by using theformula,
√[(x2 –x1)2 + (y2 – y1)2 +(z2 – z1)2]
√[(-4)2 +(-4)2 + (6)2] = √(16+16+36)
= √68
= 2√17
Now let us find thedirection cosines of the line AB
By using the formula,
-4/2√17 , -4/2√17,6/2√17
Or -2/√17, -2/√17,3/√17
Similarly,
Let us find thedirection ratios of BC
Where, B = (-1, 1, 2)and C = (-5, -5, -2)
Ratio of AB = [(x2 –x1)2, (y2 – y1)2,(z2 – z1)2]
= (-5+1), (-5-1),(-2-2) = -4, -6, -4
Then by using theformula,
√[(x2 –x1)2 + (y2 – y1)2 +(z2 – z1)2]
√[(-4)2 +(-6)2 + (-4)2] = √(16+36+16)
= √68
= 2√17
Now let us find thedirection cosines of the line AB
By using the formula,
-4/2√17, -6/2√17,-4/2√17
Or -2/√17, -3/√17,-2/√17
Similarly,
Let us find thedirection ratios of CA
Where, C = (-5, -5,-2) and A = (3, 5, -4)
Ratio of AB = [(x2 –x1)2, (y2 – y1)2,(z2 – z1)2]
= (3+5), (5+5), (-4+2)= 8, 10, -2
Then by using theformula,
√[(x2 –x1)2 + (y2 – y1)2 +(z2 – z1)2]
√[(8)2 +(10)2 + (-2)2] = √(64+100+4)
= √168
= 2√42
Now let us find thedirection cosines of the line AB
By using the formula,
8/2√42, 10/2√42,-2/2√42
Or 4/√42, 5/√42,-1/√42