Three Dimensional Geometry quiz

The direction cosines of the x-axis are:

If l, m, n are direction cosines of a line, then l^2 + m^2 + n^2 equals:

The direction cosines of a line equally inclined to the three axes are:

The angle between lines with direction ratios 1, 1, 0 and 0, 1, 1 is:

Two lines with direction ratios a1, b1, c1 and a2, b2, c2 are perpendicular if:

The shortest distance between two intersecting lines is:

The direction ratios of the line (x - 1)/2 = (y + 3)/(-1) = z/4 are:

Two lines are parallel if their direction ratios are:

The point on the line r = (i + j) + lambda(2i - j + k) corresponding to lambda = 0 is:

Skew lines are lines that are:

If a line makes angles 90, 60 and 30 degrees with the x, y and z axes, find its direction cosines. (2 marks)

Find the Cartesian equation of the line through (1, 2, 3) with direction ratios 2, -1, 4. (2 marks)

Convert the Cartesian equation (x - 5)/3 = (y + 4)/7 = (z - 6)/2 into vector form. (2 marks)

Show that the line through (1, -1, 2) and (3, 4, -2) is perpendicular to the line through (0, 3, 2) and (3, 5, 6). (2 marks)

Find the direction cosines of the vector 2i + 2j - k. (2 marks)

Can 1/2, 1/2, 1/2 be the direction cosines of a line? Justify. (2 marks)

Find the vector equation of the line through the origin and (5, -2, 3). (2 marks)

Write the formula for the shortest distance between two skew lines in vector form. (2 marks)

Find the angle between the lines with direction ratios 2, 3, 6 and 1, 2, 2. (2 marks)

Find the coordinates of a point on the line (x - 2)/1 = (y + 1)/2 = (z - 3)/(-2) at a distance of 6 units from (2, -1, 3). (2 marks)

Find the direction cosines of the line through (1, 2, 3) and (3, 4, 4). (3 marks)

Find the angle between the lines r = 2i - 5j + k + lambda(3i + 2j + 6k) and r = 7i - 6k + mu(i + 2j + 2k). (3 marks)

Find the Cartesian equation of the line through (-2, 4, -5) and parallel to the line (x + 3)/3 = (y - 4)/5 = (z + 8)/6. (3 marks)

Find the shortest distance between the lines (x + 1)/7 = (y + 1)/(-6) = (z + 1)/1 and (x - 3)/1 = (y - 5)/(-2) = (z - 7)/1. (3 marks)

A line through (1, 0, -1) has direction ratios 2, -1, 2. Find the point on it with lambda = 2 in r = a + lambda b, and its distance from (1, 0, -1). (3 marks)

Read the passage and answer the questions. A drone takes off from point (1, 2, 3) and flies in a straight line in the direction 2i + 3j + 6k (units in metres). (i) Write the vector equation of its path. (ii) Find the direction cosines of the path. (iii) Find the position after it has travelled 14 m. (5 marks)

Read the passage and answer the questions. Two pipelines in a factory follow the lines r = (i + j) + lambda(2i - j + k) and r = (2i + j - k) + mu(3i - 5j + 2k). An engineer wants to connect them with the shortest pipe. (i) Show that the lines are not parallel. (ii) Find b1 x b2. (iii) Find the length of the shortest connecting pipe. (5 marks)

Read the passage and answer the questions. Two steel rods in a frame lie along (x - 5)/7 = (y + 2)/(-5) = z/1 and x/1 = y/2 = z/3. (i) Write the direction ratios of both rods. (ii) Are the rods perpendicular? (iii) Find the angle between them. (5 marks)

Read the passage and answer the questions. Three lamp posts are located at points (1, 2, 3), (2, 4, 5) and (4, 8, 9). (i) Find the direction ratios of the segments joining the first two and the last two posts. (ii) Are the posts collinear? (iii) Write the equation of the line through them. (5 marks)

Read the passage and answer the questions. A rod makes angles of 90 degrees, 135 degrees and 45 degrees with the x, y and z axes. (i) Find its direction cosines. (ii) Verify that l^2 + m^2 + n^2 = 1. (iii) Write the equation of the rod if it passes through the origin. (5 marks)

Derive the vector and Cartesian equations of a line through two given points. Hence find the equations of the line through (3, -2, -5) and (3, -2, 6). (6 marks)

Derive the formula for the shortest distance between two skew lines with a diagram, and use it for the lines in question 24. (6 marks)

Find the foot of the perpendicular from the point (1, 2, 3) to the line (x - 6)/3 = (y - 7)/2 = (z - 7)/(-2) and the perpendicular distance. (6 marks)

Show that the lines (x - 1)/2 = (y - 2)/3 = (z - 3)/4 and (x - 4)/5 = (y - 1)/2 = z/1 intersect, and find their point of intersection. (6 marks)

Find the angle between the lines with direction ratios 1, 1, 2 and sqrt(3) - 1, -sqrt(3) - 1, 4. Draw a sketch showing the two directions. (6 marks)