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Math Help - Minimum value

  1. #1
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    Minimum value

    I'm amazed I'm stuck on this question.

    I need to find the value of t at which it becomes the minimum value.

    |\vec b|^2-2t\vec ab+t^2|\vec a|^2

    I thought you complete the square to find the minimum value. I can't get what they're getting by completing the square (although I get it when I use calculus)

    |\vec a|^2(t-\frac {\vec a*\vec b} {|\vec a|^2})^2+|\vec b|^2-\frac {(\vec a * \vec b)}{|\vec a|^2})

    How do you get that?
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  2. #2
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    Quote Originally Posted by chengbin View Post
    I'm amazed I'm stuck on this question.

    I need to find the value of t at which it becomes the minimum value.

    |\vec b|^2-2t\vec ab+t^2|\vec a|^2

    I thought you complete the square to find the minimum value. I can't get what they're getting by completing the square (although I get it when I use calculus)

    |\vec a|^2(t-\frac {\vec a*\vec b} {|\vec a|^2})^2+|\vec b|^2-\frac {(\vec a * \vec b)}{|\vec a|^2})

    How do you get that?
    You have:

    a^2 t^2 - 2c t + b^2 where c = \vec{a}\cdot \vec{b}

    = a^2 \left(t^2 - \frac{2c}{a^2} t + \frac{b^2}{a^2}\right)

    = a^2 \left( \left [t - \frac{c}{a^2}\right]^2 - \frac{c^2}{a^4} + \frac{b^2}{a^2}\right)

    = a^2 \left [t - \frac{c}{a^2}\right]^2 - \frac{c^2}{a^2} + b^2.
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