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Math Help - Vectors (Dot product?)

  1. #1
    Mtl
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    Vectors (Dot product?)

    Given that the two vectors u=(2a+b) and v=(4a-3b) are perpendicular and that |a|=3 and |b|=6, then find the angle between a and b.

    So I pretty sure i need to set the dot product of u and v = to zero. Then there is probably some sort of way to rearange it, of which I am not sure.
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    Quote Originally Posted by Mtl View Post
    Given that the two vectors u=(2a+b) and v=(4a-3b) are perpendicular and that |a|=3 and |b|=6, then find the angle between a and b.

    So I pretty sure i need to set the dot product of u and v = to zero. Then there is probably some sort of way to rearange it, of which I am not sure.
    Hello,

    the definition of the dot product says:

    \vec a \cdot \vec b = |\vec a| \cdot |\vec b| \cdot \cos(\theta) and therefore:

    \cos(\theta)=\frac{\vec a \cdot \vec b}{|\vec a| \cdot |\vec b|}

    From your problem you know:

    (2\vec a + \vec b) \cdot (4 \vec a - 3\vec b) = 0..... Expand the brackets:

    8(\vec a)^2 - 6\vec a \vec b + 4\vec a \vec b -3(\vec b)^2 = 0

    Since (\vec a)^2 = 9 and (\vec b)^2 = 36

    the equation becomes:

    -2(\vec a \cdot \vec b) = 36 and therefore \vec a \cdot \vec b = -18

    Now plug in all values into the formula to calculate the angle:

    \cos(\theta) = \frac{-18}{3 \cdot 6}=-1~\implies~\theta = 180^\circ
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  3. #3
    Mtl
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    Thanks, well done in explaining the solution.
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    Quote Originally Posted by Mtl View Post
    Thanks, well done in explaining the solution.
    Hi,

    thanks for your kind reply BUT: your problem is a little bit more tricky than it seems to be:

    1. According to my result the vectors \vec a and \vec b are pointing in opposite directions, that means they are collinear. And therefore any linear combinations of \vec a and \vec b must be parallel. Seems to be a contradiction to the given conditions.

    2. Considering the directions of \vec a and \vec b and the length of \vec a and \vec b then

    \vec u = 0

    and everything is nice and peaceful again.
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