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4 problems match “direction cosines”. Clear

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1. Angle with the z-axis ●●○○○ 20 ptsA vector makes an angle of $\frac{\pi}{4}$ with each of the x-axis and the y-axis. Find the angle, in degrees, that the vector makes with the z-axis.
2. Angle with z-axis from two axis angles ●●○○○ 20 ptsA line makes an angle of $60^\circ$ with each of the $x$-axis and the $y$-axis. Find the acute angle, in degrees, that the line makes with the $z$-axis.
3. Angles of a unit vector with the axes ●●○○○ 20 ptsA unit vector $\vec a$ makes an angle $\frac{\pi}{3}$ with $\hat i$, an angle $\frac{\pi}{4}$ with $\hat j$, and an acute angle $\theta$ with $\hat k$. Find $\theta$ in degrees, and hence find the components of $\vec a$ along $\hat i$, $\hat j$, $\hat k$, in that order. Round the second component to two decimal places.
4. Angle between lines from direction cosines ●●●○○ 50 ptsTwo lines have direction cosines $(l_1,m_1,n_1)$ and $(l_2,m_2,n_2)$. Each line satisfies $l+m+n=0$ and $\frac{mn}{q-r}+\frac{nl}{r-p}+\frac{lm}{p-q}=0$, where $p$, $q$, $r$ are distinct constants. Find the acute angle between the two lines, in degrees.