The direction ratios of a line are determined by:

a) Ratios of its equation coefficients

b) Components of its direction vector

c) Ratios of its coordinates

d) Components of its equation

How is the shortest distance between two skew lines typically determined?

a) Using vector addition.

b) Through scalar triple product.

c) Via vector cross product.

d) With vector dot product.

Skew lines are lines that:

a) Are parallel but do not intersect.

b) Lie in the same plane but do not intersect.

c) Are not coplanar and do not intersect.

d) Intersect at a right angle.

The direction cosines of a line are given by:

a) Ratios of its coordinates

b) Components of its equation

c) Ratios of its coefficients

d) Components of its direction vector

Which equation represents the Cartesian equation of a line passing through points \( P(1,2,3) \) and \( Q(4,5,6) \)?

a) \( \frac{x-1}{3} = \frac{y-2}{3} = \frac{z-3}{3} \)

b) \( \frac{x-1}{4} = \frac{y-2}{5} = \frac{z-3}{6} \)

c) \( \frac{x-1}{3} = \frac{y-2}{3} = \frac{z-3}{4} \)

d) \( \frac{x-1}{4} = \frac{y-2}{5} = \frac{z-3}{6} \)

If the direction cosines of a line are \( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \), what can be concluded about the line?

a) The line is parallel to the x-axis.

b) The line is parallel to the y-axis.

c) The line is parallel to the z-axis.

d) The line passes through the origin.

Skew lines are:

a) Parallel lines

b) Coplanar lines

c) Intersecting lines

d) Non-intersecting and non-parallel lines

What is the formula for finding the angle between two lines with direction cosines \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \)?

a) \( \cos^{-1}(l_1 l_2 + m_1 m_2 + n_1 n_2) \)

b) \( \cos^{-1}(l_1 + m_1 + n_1 + l_2 + m_2 + n_2) \)

c) \( \cos^{-1}\left(\frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \sqrt{l_2^2 + m_2^2 + n_2^2}}\right) \)

d) \( \cos^{-1}\left(\frac{l_1 + m_1 + n_1}{l_2 + m_2 + n_2}\right) \)

Which equation represents the vector equation of a line passing through points \( P(1,1,1) \) and \( Q(2,2,2) \)?

a) \( r = \langle 1,1,1 \rangle + \lambda \langle 1,1,1 \rangle \)

b) \( r = \langle 1,1,1 \rangle + \lambda \langle 2,2,2 \rangle \)

c) \( r = \langle 1,1,1 \rangle + \lambda \langle 1,2,1 \rangle \)

d) \( r = \langle 1,1,1 \rangle + \lambda \langle 1,1,2 \rangle \)