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Homework answers / question archive / Find, correct to the nearest degree, the three angles of the triangle with the given vertices
Find, correct to the nearest degree, the three angles of the triangle with the given vertices.
A(1, 0, −1), B(5, −3, 0), C(1, 5, 2)
∠CAB =
| ∠ABC = | |
| ∠BCA = |
AB = (5,-3,0) - (1,0,-1) = (4,-3,1)
AC= (1,5,2) - (1,0,-1) = (0,5,3)
|AB| = sqrt (4^2+3^2+1^2) = sqrt(26)
|AC| = sqrt (0 + 5^2+3^2) = sqrt(34)
use:
AB.AC = |AB|*|AC| cos ∠CAB
(4,-3,1).(0,5,3) = sqrt(26)*sqrt(34)* cos ∠CAB
4*0 -3*5 +1*3 = sqrt(26)*sqrt(34)* cos ∠CAB
-12 = sqrt(26)*sqrt(34)* cos ∠CAB
cos ∠CAB = - 0.404
∠CAB = 113.8 degree
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BA = (1,0,-1) - (5,-3,0) = (-4,3,-1)
BC= (1,5,2)-(5,-3,0) = (-4,8,2)
|BA| = sqrt(26)
|BC| = sqrt (4^2 + 8^2 + 2^2) = sqrt(84)
use:
BA.BC = |BA|*|BC|* cos ∠ABC
(-4,3,-1).(-4,8,2) = sqrt(26) * sqrt(84) * cos ∠ABC
16+24-2 = sqrt(26) * sqrt(84) * cos ∠ABC
cos ∠ABC -= 0.813
∠ABC = 35.6 degree
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sum of all angle = 180
∠BCA + 35.6 + 113.8 = 180
∠BCA = 30.6 degree
