Three Dimensional Geometry Quiz

What are direction cosines of a line?

• The slopes of the line in three-dimensional space

• The ratios of the coordinates of the line

• The lengths of the line segments

• The angles made by the line with the coordinate axes

If a line makes angles 90°, 60°, and 30° with the x, y, and z-axes respectively, what are its direction cosines?

• 0, 1/2, √3/2

• 1, 0, 0

• 1/√2, 1/√2, 0

• 1, 1, 1

What is the relationship between direction cosines and direction ratios?

• Direction cosines are always integers

• Direction cosines can be negative only

• Direction ratios are proportional to direction cosines

• Direction ratios are always positive

What is the formula for the angle between two lines with direction ratios a1, b1, c1 and a2, b2, c2?

• cosθ = (a1a2 + b1b2 + c1c2) / (√(a1² + b1² + c1²)√(a2² + b2² + c2²))

• sinθ = (a1 + b1 + c1) / (a2 + b2 + c2)

• tanθ = (b1 / a1) + (b2 / a2)

• θ = a1 + b1 + c1 + a2 + b2 + c2

What is the equation of a line through a point (x1, y1, z1) with direction cosines l, m, n?

• x - x1 / l = y - y1 / m = z - z1 / n

• x + x1 = l(y + y1) + m(z + z1)

• x = l + x1, y = m + y1, z = n + z1

• x² + y² + z² = l² + m² + n²

What is the shortest distance between two skew lines?

• The sum of their lengths

• The length of the perpendicular drawn from one line to the other

• The distance between their direction ratios

• The distance between their direction cosines

If two lines are parallel, what can be said about their direction ratios?

• They are perpendicular

• They are independent

• They are equal

• They are proportional

What is the formula for the direction cosines of a line segment joining two points P(x1, y1, z1) and Q(x2, y2, z2)?

• PQ / (x2 - x1), PQ / (y2 - y1), PQ / (z2 - z1)

• (x1 - x2) / PQ, (y1 - y2) / PQ, (z1 - z2) / PQ

• (x2 - x1) / PQ, (y2 - y1) / PQ, (z2 - z1) / PQ

• (x2 + x1) / PQ, (y2 + y1) / PQ, (z2 + z1) / PQ

What is the condition for two lines to be perpendicular in terms of their direction ratios?

• a1 + a2 + b1 + b2 + c1 + c2 = 0

• a1a2 + b1b2 + c1c2 = 0

• a1² + b1² + c1² = a2² + b2² + c2²

• a1a2 = b1b2 = c1c2

What is the vector equation of a line passing through a point A and parallel to a vector b?

• r = a + b

• r = λa + b

• r = a - λb

• r = a + λb