List of top Mathematics Questions on Direction Cosines and Direction Ratios of a Line asked in CBSE CLASS XII

A unit vector $\vec{a}$ is such that it makes an angle $\frac{\pi}{4}$ with the $x$-axis, $\frac{\pi}{3}$ with the $y$-axis and an acute angle $\theta$ with the $z$-axis. Find $\theta$ and the components of $\vec{a}$.

If \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \) are direction cosines of lines \( L_1 \) and \( L_2 \) respectively and \( \theta \) is the acute angle between them, then :

Direction ratios of lines \( l_1 \) and \( l_2 \) respectively are \( \langle 1, -2, 3 \rangle \) and \( \langle -2, p, -6 \rangle \). The value of p for which \( l_1 \parallel l_2 \), is :

Assertion (A): A line can have direction cosines $\langle 1, 1, 1 \rangle$.
Reason (R): $\cos\theta = 1$ is possible for $\theta = 0$.

The direction cosines of the line given by $x = y = 1 - z$ are:

Assertion (A): A line can have direction cosines $\langle 1,1,1\rangle$.
Reason (R): $\cos\theta=1$ is possible for $\theta=0^\circ$.

Direction cosines of line $x=y=1-z$ are