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:

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

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

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

a) Parallel.

b) Perpendicular.

c) Skew.

d) Intersecting.

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.

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.