Question Bank: Three Dimensional Geometry

What are the direction cosines of a line?

If a line makes angles α, β, and γ with the x, y, and z axes, respectively, what is true about the direction cosines l, m, n?

What do direction ratios represent for a line in three-dimensional space?

What happens to the direction cosines if the direction of a line is reversed?

If a line does not pass through the origin, how can we determine its direction cosines?

Which set of factors can describe the direction ratios of a line if l, m, n are its direction cosines?

What conclusion can we draw if the direction ratios of a line are all zero?

How are direction cosines typically denoted for a line in space?

What geometric rule describes the relationship between direction cosines?

Which of the following equations represents the standard form of a line in 3D space using direction ratios?

How many different sets of direction cosines can a single directed line have?

The direction ratios of a line are proportional to 2, 3, and 4. What are possible direction cosines for this line?

If two lines in 3D space are parallel, how do their direction ratios behave?

What can be concluded about two lines if their direction cosines satisfy l1/l2 = m1/m2 = n1/n2?

What are the direction cosines of the x-axis?

If a line makes angles 90°, 135°, and 45° with the x, y, and z axes respectively, what are its direction cosines?

How do you determine if points A(2, 3, -4), B(1, -2, 3), and C(3, 8, -11) are collinear?

What is the Cartesian equation of a line passing through point (1, 2, 3) with direction ratios 4, -4, 2?

If a line in space passes through points P(1, 2, 3) and Q(4, -1, 5), what is the direction vector?

Which of the following conditions uniquely determines a line in space?

Find the direction cosines if a line has direction ratios -18, 12, -4.

What is the vector equation of a line parallel to the vector (2, -3, 1) passing through point (1, 2, 3)?

How can you express the vector equation of a line in symmetric form?

In general, what is the direction cosine for an angle of 180°?

What condition indicates two lines are skew lines?

What geometric figure is formed by a line in 3D that is defined by two distinct points?

If two points in space are (1, 2, 3) and (4, 5, 6), what is the vector from the first point to the second?

What does the term 'direction ratios' refer to in a 3D line?

In the line equation r = a + tb, what does 'b' represent?

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

If the direction ratios of two lines are proportional, what can be inferred about the angle between them?

For two lines represented by direction ratios (2, -1, 3) and (4, 2, 6), what is the cosine of the angle between them?

What is the acute angle between the lines with direction cosines (1/√3, 1/√3, 1/√3) and (0, 1, 0)?

If two lines with direction ratios (1, 2, 3) and (2, 4, 6) are given, what type of lines are they?

Which of the following pairs of lines are perpendicular?

What are the direction cosines of a line that makes angles of 60°, 45°, and 30° with the x, y, and z axes, respectively?

How do you express the angle between two lines with given direction cosines?

If the direction cosines of a line are l = 0.6, m = 0.8, what is the direction cosine n?

Given lines with direction ratios (-1, 2, 1) and (4, -8, 4), what can be said about their relationship?

Which of the following is true about direction ratios (a, b, c) and direction cosines (l, m, n)?

What is the sine of an angle θ if cos θ = 0.5?

If a line has direction ratios of 2, 3, and 6, what are its direction cosines?

When is the sine of the angle between two lines equal to 1?

What is the relationship between the direction cosines of a line and the angle it makes with the axes?

For lines with direction cosines (1, 0, 0) and (0, 1, 0), find cos θ.

Which direction ratios correspond to the direction cosines of -1/√3, 1/√3, 1/√3?

What would be the angle θ if θ = 45° and you switch the direction of one line?

In the equation of a line, if the direction cosines are known, how can the direction ratios be determined?

How is a line's direction changed when its direction cosines are reversed?

If a line has direction ratios a, b, c, what value of k is derived from the formula l² + m² + n² = 1?

What is true when you find direction cosines and they do not add up to 1?

Given direction cosines l = 1/√2, m = 1/√2, what are the corresponding angles of the line with the axes?

Which of the following describes the relationship between three direction cosines when their squares are summed?

If λ is a constant that connects direction ratios and direction cosines, how can two sets of ratios relate?

What is the solution for direction ratios when given cosines like 0.5, 0.5, 0.707?

What is the shortest distance between two intersecting lines?

If two lines are parallel, how do we find their shortest distance?

Which of the following describes skew lines?

What is the formula for calculating the shortest distance between two skew lines represented by vectors?

If the direction ratios of two skew lines are (1, 2, 3) and (4, 5, 6), what is their cross product?

The shortest distance between two skew lines can be represented as?

Which of the following statements is NOT true about skew lines?

Calculate the shortest distance between the lines given by the equations: r1 = (1, 2, 3) + λ(4, 3, 2) and r2 = (0, 0, 0) + μ(1, 1, 1).

For the pair of lines l1: (1,1,1) + λ(1,2,3) and l2: (2,3,4) + μ(4,5,6), what type of lines are they?

If two lines L1 and L2 have direction ratios (2, 3, 5) and (4, 6, 9) respectively, are they coplanar?

What angle must the vector for the shortest distance between two skew lines be at?

Given the direction ratios (1, 2, 2) and (4, 5, 6) of two lines, how would you determine the shortest distance?

Identify the primary characteristic of the vector representing the shortest distance between two skew lines.

How can the distance between skew lines be visually represented in three-dimensional space?