The shortest distance between two skew lines is given by:

a) The length of a perpendicular segment between the lines

b) The magnitude of the cross product of their direction vectors

c) The magnitude of the dot product of their direction vectors

d) The difference in their direction ratios

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.

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

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

If two lines are perpendicular, what can be said about their direction ratios?

a) Their dot product is zero.

b) Their cross product is zero.

c) The sum of their direction ratios is zero.

d) The difference of their direction ratios is zero.

What do the direction cosines of a line indicate?

a) The angles between the line and the coordinate axes.

b) The ratios of the direction components of the line.

c) The perpendicular distances from the line to the coordinate axes.

d) The inclinations of the line with the coordinate axes.

Two lines with direction ratios \( 2, 1, -1 \) and \( 3, -1, 2 \) are:

a) Parallel.

b) Perpendicular.

c) Skew.

d) Intersecting.

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