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

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.

Skew lines are:

a) Parallel lines

b) Coplanar lines

c) Intersecting lines

d) Non-intersecting and non-parallel lines

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.

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.

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

a) Parallel.

b) Perpendicular.

c) Skew.

d) Intersecting.

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

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