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if vectors a+2b and 5a-4b are perpendicular to each other and a and b are unit vectors. Find the angle between a and b.

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  1. Let θ be the angle between a and b. We know that cosθ = (a∙b) / (|a||b|). Since a and b are unit vectors, |a|=|b|=1. Therefore we need to find a∙b.

    a+2b and 5a-4b are perpendicular, so

    (a+2b)∙(5a-4b)=0

    Multiply this out to obtain

    5a∙a + 6a∙b - 8b∙b = 0

    which results in

    6a∙b = 8b∙b - 5a∙a

    Since a and b are unit vectors,

    6a∙b = 8-5 = 3

    Therefore, cos θ = 3/6. θ=π/3

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  2. Good!

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  3. Good

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  4. If two vectors are perpendicular their dot product will equal zero.
    (a+2b)•(5a-4b)=0
    5a•a+10a•b-4a•b-8b•b=0
    6a•b=8b•b-5a•a
    6a•b=8-5=3
    cosθ=3/6.
    θ=π/3

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