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Math Help - Projection of a vector onto a line

  1. #1
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    Smile Projection of a vector onto a line

    I need to find the projection of vector \vec{a} = \sqrt{2}\cdot\vec{i}-3\cdot\vec{j}-5\cdot\vec{k} onto the line that makes \alpha = 45^{\circ} angle with Ox, \gamma = 60^{\circ} angle with Oz and an acute angle with Oy
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  2. #2
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    Quote Originally Posted by sillyme View Post
    I need to find the projection of vector \vec{a} = \sqrt{2}\cdot\vec{i}-3\cdot\vec{j}-5\cdot\vec{k} onto the line that makes \alpha = 45^{\circ} angle with Ox, \gamma = 60^{\circ} angle with Oz and an acute angle with Oy
    If a line makes angles \alpha, \beta, \gamma with the x, y, and z axes respectively, then the "direction cosines" satisfy cos^2(\alpha)+ cos^2(\beta)+ cos^2(\gamma)= 1 and the vector cos(\alpha)\vec{i}+ cos(\beta)\vec{j}+ cos(\gamma)\vec{k} is a unit vector in the direction of the line.

    Here, \alpha= 45 so cos(\alpha)= \frac{1}{\sqrt{2}} and cos^2(\alpha)= \frac{1}{2}.

    \beta= 60 so cos(\beta)= \frac{1}{2} and cos^2(\beta)= \frac{1}{4}.

    That means that cos^2(\gamma)= 1- 1/2- 1/4= 1/4 and so cos(\gamma)= 1/2

    A unit vector in the direction of the line is \frac{\sqrt{2}}{2}\vec{i}+ \frac{1}{2}\vec{j}+ \frac{1}{2}\vec{k}

    Can you find the projection of your given vector onto that?
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  3. #3
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    Yes. Now I can calculate the projection. Thank you HallsofIvy!
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