Three Dimensional Geometry — Practice

Question 1

Find the direction cosines of a line making equal angles with all three axes.

Accepted Answer

If l = m = n then 3l^2 = 1, so l = +/- 1/sqrt(3). DCs are (1/sqrt3, 1/sqrt3, 1/sqrt3) or their negatives.

Question 2

Find the vector equation of the line through (1,2,3) and (4,5,6).

Accepted Answer

a = i + 2j + 3k, b = 3i + 3j + 3k. r = (i + 2j + 3k) + lambda(3i + 3j + 3k).

Question 3

Find the angle between the planes x + 2y + 2z = 9 and 2x + 3y + 6z = 8.

Accepted Answer

n1 = (1,2,2) with |n1| = 3, n2 = (2,3,6) with |n2| = 7, n1.n2 = 20. cos(theta) = 20/21, so theta = arccos(20/21).

Question 4

Find the shortest distance between r = (i+2j+k) + lambda(i-j+k) and r = (2i-j-k) + mu(2i+j+2k).

Accepted Answer

a2-a1 = i - 3j - 2k. b1 x b2 = -3i + 0j + 3k with magnitude 3 sqrt(2). (a2-a1).(b1 x b2) = -9. d = 9/(3 sqrt 2) = 3/sqrt(2) units.

Question 5

Show the lines (x-1)/2 = (y-2)/3 = (z-3)/4 and (x-4)/5 = (y-1)/2 = z are coplanar.

Accepted Answer

b1 = (2,3,4), b2 = (5,2,1), a2-a1 = (3,-1,-3). b1 x b2 = (-5,18,-11). (a2-a1).(b1 x b2) = -15 - 18 + 33 = 0, so the lines are coplanar.

Practice — Three Dimensional Geometry