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I'm not going to lie, I am terrible at physics and I had no idea how to solve this question, so I asked someone for help. I came out with a long sheet of equations I never heard of nor what was mentioned in my textbook and a final answer of (.94, .34). I don't know if that is right. I never heard of a dot product, never heard of a lot of stuff they used. Can anyone verify if this answer looks right...? (and if you have the smallest inclination, if you know a different way, to explain this, that wouldn't be bad either )

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Let vector A = 5.00i + 11.0j and vector B = 2.00i - 1.00j

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Magnitude of A = A = √52+112 = 12.08

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Magnitude of B = B = √22+12 = 2.24

Dot product of the two vectors A = Ax i + Ay j and B = Bx i + By j is given by:

A.B = ABcosθ …….(1) where θ is the angle between the two vectors

and A.B = AxBx +AyBy ………(2)

Let us first find dot product of the given vectors using (1) and (2):

Using (1) we get: A.B = ABcosθ = 12.08 x 2.24 cosθ = 27cosθ ……(3)

Using (2) we get: A.B = 5 x 2 - 11 x 1 = -1 ……..(4)

Equating (3) and (4): 27cosθ = -1 or cosθ = -1/27 = 0.037 or θ = 92O

Bisector vector will divide the angle between the given vectors into two equal angles i.e. the angle between the bisector vector and any one of the given vectors shall be 46O.

Let the bisector vector be represented by: C = Cx i + Cy j

Further, we know that the bisector vector is a unit vector. Hence, the magnitude of vector C is 1. Hence,

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√ Cx2 + Cy2 = 1 or Cx2 + Cy2 = 1 ………(5)

Further, we find the dot product between vector A and vector C using (1) and(2):

Using (1) we get: A.C = ACcos(θ/2) = 12.08 x 1 cos46O = 8.39

Using (2) we get: A.C = 5Cx + 11Cy

Equating we get: 5Cx+11Cy = 8.39 …….(6)

From (5) Cy2 = 1 - Cx2 …….(7)

From (6) Cy2 = (8.39 - 5Cx)2/112 = (8.39 - 5Cx)2/121 …….(8)

Equating (7) and (8): (8.39 - 5Cx)2/121 = 1 - Cx2

70.39 + 25Cx2 – 83.9Cx = 121 - 121Cx2

146Cx2 – 83.9Cx – 50.61 = 0

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Cx = [- (– 83.9) + √(– 83.9)2 – 4(146)( – 50.61)]/2(146)

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Cx = [+83.9 + √7039 + 29556]/292

Cx = [+83.9 + 191.3]/292

Taking + sign: Cx = 0.94

Substituting for Cx in (7): Cy2 = 1 - 0.942 = 0.1164 or Cy = 0.34

Hence, the bisector unit vector is (0.94 i + 0.34 j)