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

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.

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 parallel to the vector \( \langle 1, -2, 3 \rangle \) passing through the point \( (4,5,6) \)?

a) \( r = \langle 4,5,6 \rangle + \lambda \langle 1,-2,3 \rangle \)

b) \( r = \langle 4,5,6 \rangle + \lambda \langle -1,2,-3 \rangle \)

c) \( r = \langle 1,-2,3 \rangle + \lambda \langle 4,5,6 \rangle \)

d) \( r = \langle 4,5,6 \rangle + \lambda \langle 3,2,-1 \rangle \)

Skew lines are:

a) Parallel lines

b) Coplanar lines

c) Intersecting lines

d) Non-intersecting and non-parallel lines

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

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

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.

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.