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Math Help - Gr.12 Vector Question Help Please!?

  1. #1
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    Gr.12 Vector Question Help Please!?

    Two vectors 2a + b and a - 3b are perpendicular. Find the angle between a and b, if |a| = 2|b| .

    Step-by-step solution please and thank you!

    Been solved but feel free to try!
    Last edited by narutoblaze; February 15th 2009 at 12:02 PM.
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  2. #2
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    I was about halfway through writing my post when you said you solved it, but I finished it and I'll put it up anyway. The problem itself was easy, but typing out the \text\LaTeX was tedious:

    The vectors 2\textbf a+\textbf b and \textbf a - 3\textbf b are orthogonal, so (2\textbf a+\textbf b)\cdot(\textbf a - 3\textbf b) = 0\text,

    \Rightarrow2\textbf a\cdot\textbf a-5\textbf a\cdot\textbf b-3\textbf b\cdot\textbf b = 0

    \Rightarrow2\lVert\textbf a\rVert^2-3\lVert\textbf b\rVert^2-5\textbf a\cdot\textbf b=0

    \Rightarrow2(2\lVert\textbf b\rVert)^2-3\lVert\textbf b\rVert^2-5\textbf a\cdot\textbf b=0

    \Rightarrow5\lVert\textbf b\rVert^2=5\textbf a\cdot\textbf b\Rightarrow\lVert\textbf b\rVert^2=\textbf a\cdot\textbf b

    \Rightarrow1=\frac{\textbf a\cdot\textbf b}{\lVert\textbf b\rVert\lVert\textbf b\rVert}

    \Rightarrow1=\frac{\textbf a\cdot\textbf b}{\frac12\lVert\textbf a\rVert\lVert\textbf b\rVert}

    \Rightarrow\frac12=\frac{\textbf a\cdot\textbf b}{\lVert\textbf a\rVert\lVert\textbf b\rVert}

    \Rightarrow\cos\frac\pi3=\frac{\textbf a\cdot\textbf b}{\lVert\textbf a\rVert\lVert\textbf b\rVert}

    So the angle between \textbf a and \textbf b is \frac\pi3\text.
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