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Grade 12 Math

if a and b are unit vectors, and magnitude of vectors a +b = sq. root of 3, determine (2a-5b) dot (b+3a).

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  1. (2a-5b)•(3a+b)
    2a•3a - 13a•b - 5b•b
    6 - 13a•b - 5
    1 - 13a•b

    Now |a+b| = √3, so using the law of cosines to get the angle θ between a and b, we have

    1+1-2cosθ = √3
    cosθ = 1-√3/2

    Now, we know that since a and b are unit vectors,
    a•b = cosθ = 1-√3/2

    Now we have

    (2a-5b)•(3a+b) = 1 - 13a•b
    = 1-13(1-√3/2)
    = 13√3/2 - 12
    = -0.741

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