Q.293. Vector problem in R^2.

Question 293.
Three vectors U, V and W in R^2 are such that U + V + W = 0 and the angle between W and V is 60°. l V l = 2 and l W l = 3.
Find U and the angles between V & U, and, U & W.

Answer 293.
U + V + W = 0
=> U = - (V + W)
=> l U l^2
= l V l^2 + l W l^2 + 2 V.W
= 2^2 + 3^2 + 2 *(2) * (3) cos60°
= 19
=> U = √(19)

 W = - (U + V)
=> l W l^2 = l U l^2 + l V l^2 + 2 U.V
=> 3^2 = [√(19)]^2 + (2)^2 + 2 * [√(19)] * 2 cos(V^U)
=> cos(V^U) = - (14) / [4√(19)]
=> angle between V & U = 143.4°

V = - (U + W)
=> l V l^2 = l U l^2 + l W l^2 + 2 U.W
=> 2^2 = [√(19)]^2 + (3)^2 + 2 * [√(19)] * 3 cos(U^W)
=> cos(U^W) = - (24) / [6√(19)]
=> angle between U & W = 156.6°.

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