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A line $OP$ is inclined at $45^o$ to the $x$-axis and $120^o$ to the $y$-axis. Find its inclination to the $z$-axis.

I think the inclination to the $y$-axis does not affect inclination to the $z$-axis, so the answer is $45^o$. However, I don't know how to use dot product to do calculation.

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    If you don’t think that the $y$-inclination affects $z$, then why should the $x$-inclination affect it, either? – 2017-01-13

2 Answers 2

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Hint: Let $\mathbf u=(u_x,u_y,u_z)$ be a unit direction vector for the line, so that $\|\mathbf u\|=\sqrt{\mathbf u\cdot\mathbf u}=\sqrt{u_x^2+u_y^2+u_z^2}=1$. You have $\mathbf u\cdot\mathbf e_x=u_x=\cos{45°}$ and $\mathbf u\cdot\mathbf e_y=u_y=\cos{120°}$. Solve for $u_z$, which will also be the cosine of the inclination from the $z$-axis $\mathbf u\cdot\mathbf e_z$.

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$\cos^2(45) + \cos^2(120)+\cos^2(\alpha)=1$

$\frac{1}{\sqrt2} + \frac{1}{2^2} + \cos^2(\alpha)=1$

$\cos^2(\alpha)=?$

I think you can do it from here