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Math Help - Dot product Question

  1. #1
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    Dot product Question

    The vectors x = (-4, p, -2) and y = (-2, 3, 6) are such that \arccos(\frac{4}{21}) = \theta, where \theta is the angle between x and y. Determine the value(s) of p.

    I know that x • y must equal to 4 in this case, and the product between the magnitudes of x and y must equal to 21,
    but how do you set it up so that both conditions will be true?
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  2. #2
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    Quote Originally Posted by carnage View Post
    The vectors x = (-4, p, -2) and y = (-2, 3, 6) are such that \arccos(\frac{4}{21}) = \theta, where \theta is the angle between x and y. Determine the value(s) of p.

    I know that x • y must equal to 4 in this case, and the product between the magnitudes of x and y must equal to 21,
    but how do you set it up so that both conditions will be true?
    cos\theta = \frac{4}{21}

    The dot product is a.b = |a||b|cos\theta

    (-4\cdot -2) + (p \cdot 3) + (-2 \cdot 6) = \sqrt{(-4)^2 + p^2+(-2)^2} \times \sqrt{(-2)^2+3^2+6^2} \times \frac{4}{21}


    \rightarrow 8 + 3p - 12 = \sqrt{20+p^2} \times \sqrt{49} \times \frac{4}{21}

    \rightarrow 3p - 4 = 7\sqrt{20+p^2} \times \frac{4}{21} = \frac{4\sqrt{20+p^2}}{3}

    Solve for p
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