Three Dimensional Geometry Class 12 Mcq Test (121101)

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.

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

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

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

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 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

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.

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