Three Dimensional Geometry Class 12 Mcq Test (121101)

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:

a) Parallel lines

b) Coplanar lines

c) Intersecting lines

d) Non-intersecting and non-parallel lines

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

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 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.

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

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.

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.