SOLUTION: find, correct to the nearest degree the three angles of a triange with vertices A(0,3,3) B(-2,1,1) C(1,5,-1)

Algebra ->  Triangles -> SOLUTION: find, correct to the nearest degree the three angles of a triange with vertices A(0,3,3) B(-2,1,1) C(1,5,-1)      Log On


   

Question 387508: find, correct to the nearest degree the three angles of a triange with vertices A(0,3,3) B(-2,1,1) C(1,5,-1)
Answer by Edwin McCravy(20081) About Me  (Show Source):
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find, correct to the nearest degree the three angles of a triange with vertices A(0,3,3) B(-2,1,1) C(1,5,-1)
The line segment AB is parallel to the vector < -2-0,1-3,1-3 > = < -2,-2,-2 >

The line segment AC is parallel to the vector < 1-0,5-3,-1-3 > = <  1, 2,-4 >
 
The line segment BC is parallel to the vector < 1-(-2),5-1,-1-1 > = <  3, 4,-2 >

We use this formula

cos%28theta%29%22%22=%22%22u%2Av%2F%28abs%28abs%28u%29%29abs%28abs%28v%29%29%29

Angle A is the angle between AB and AC, which is the same angle
as the angle between the vectors < -2,-2,-2 > and <  1, 2,-4 >

Their dot product is 

< -2,-2,-2 >@<  1, 2,-4 > = (-2)(1)+(-2)(2)+(-2)(-4) = -2-4+8 = 2

The norm of < -2,-2,-2 > is sqrt%28%28-2%29%5E2%2B%28-2%29%5E2%2B%28-2%29%5E2%29=sqrt%284%2B4%2B4%29=sqrt%2812%29 

The norm of < 1,2,-4 > is sqrt%28%281%29%5E2%2B%282%29%5E2%2B%28-4%29%5E2%29=sqrt%281%2B4%2B16%29=sqrt%2821%29

Substituting in the formula for the cosine,

cos%28A%29%22%22=%22%222%2F%28sqrt%2812%29sqrt%2821%29%29=.1259881577

So A = 82.76217579° or 83° to the nearest degree.

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Find the other two angles the same way.

Edwin