Worksheet: Angle between Two Straight Lines in Three Dimensions

In this worksheet, we will practice finding the angle between two straight lines in three dimensions.

Q1:

A straight line 𝐿 passes through the two points 𝐴 ( 2 , 2 , 3 ) and 𝐵 ( 6 , 4 , 5 ) , and a straight line 𝐿 passes through the two points 𝐶 ( 1 , 4 , 1 ) and 𝐷 ( 9 , 6 , 9 ) . Find the measure of the angle between the two lines, giving your answer to two decimal places if necessary.

Q2:

Find, to the nearest second, the measure of the angle between the straight line 𝑥 + 1 2 = 𝑦 2 4 = 𝑧 + 2 5 and the positive direction of the 𝑥 -axis.

  • A 7 5 2 1 2
  • B 6 2 2 4 6
  • C 8 3 5 6 5 7
  • D 7 2 3 9 1 4

Q3:

Determine, to the nearest second, the measure of the angle between the two lines that have direction ratios of ( 4 , 3 , 4 ) and ( 3 , 3 , 1 ) .

  • A 8 8 9 3 9
  • B 6 3 3 6 3
  • C 8 7 5 6 4 8
  • D 2 6 2 3 5 7

Q4:

Find the measure of the angle between the straight line 𝑥 = 1 , 𝑦 = 2 and the straight line 𝑦 = 1 , 𝑧 = 0 .

Q5:

Find, to the nearest second, the measure of the angle between the two straight lines 2 𝑥 = 4 𝑦 = 3 𝑧 and 4 𝑥 = 5 𝑦 = 2 𝑧 .

  • A 6 0 6 5 1
  • B 1 3 4 3 2 1
  • C 2 9 5 3 9
  • D 7 6 1 6 3 9

Q6:

Find the measure of the angle between the two straight lines 𝐿 𝑥 = 5 8 𝑡 , 𝑦 = 3 4 𝑡 , 𝑧 = 5 + 6 𝑡 and 𝐿 𝑥 5 3 = 𝑦 + 5 6 = 𝑧 2 2 , and round it to the nearest second.

  • A 9 9 9 3 0
  • B 4 7 3 1 2 5
  • C 8 1 3 6 3 7
  • D 8 0 5 0 2 9

Q7:

Find the measure of the angle between the two straight lines whose direction cosines are 3 1 3 2 , 9 1 1 2 , 3 7 2 and 1 0 1 3 2 , 8 1 3 2 , 9 8 2 . Give your answer to the nearest second.

  • A 6 0 2 8 2 6
  • B 1 2 5 3 3 4 1
  • C 7 0 4 4 2 9
  • D 5 4 2 6 1 8

Q8:

Find the measure of the angle between the two straight lines 𝑟 = 2 7 , 2 3 , 1 + 𝑡 2 7 , 4 3 , 9 5 and 6 𝑥 2 7 = 4 𝑦 3 6 = 3 8 𝑧 5 .

  • A 1 3 9 5 7 2 6
  • B 3 1 5 3 4 8
  • C 2 0 1 4
  • D 4 0 2 3 3

Q9:

If 𝐿 𝑥 = 0 1 , 𝑦 = 𝑧 and 𝐿 𝑦 = 0 2 , 𝑥 = 𝑧 , then find the value of 𝜃 .

Q10:

Given that a straight line passes through the origin and the point ( 4 , 1 , 2 ) , determine the exact value of c o s 𝜃 .

Note that 𝜃 is the measure of the angle between the straight line and the positive direction of the 𝑧 -axis.

  • A 2 1 2 1
  • B 4 2 1 2 1
  • C 4 2 1 2 1
  • D 2 2 1 2 1

Q11:

If a straight line makes direction angles of measures 6 0 with the 𝑦 -axis and 6 0 with the 𝑧 -axis, then find the direction angle it makes with the 𝑥 -axis.

Q12:

Find, to the nearest second, the measure of the angle between the straight line 𝑥 + 1 2 = 𝑦 1 3 = 𝑧 + 4 4 and the positive direction of the 𝑥 -axis.

  • A 7 1 1 4 1 8
  • B 2 3 1 1 5 5
  • C 8 2 2 7 1 8
  • D 6 8 1 1 5 5

Q13:

Find, to the nearest second, the measure of the angle between the straight line 𝑥 + 5 2 = 𝑦 1 4 = 𝑧 2 1 and the positive direction of the 𝑥 -axis.

  • A 5 0 5 7 1 5
  • B 7 1 2 4 5 0
  • C 6 2 2 5 3 0
  • D 6 4 7 2 4

Q14:

Find, to the nearest second, the measure of the angle between the two straight lines 5 𝑥 = 𝑦 = 3 𝑧 and 3 𝑥 = 3 𝑦 = 4 𝑧 .

  • A 4 5 5 4 5 4
  • B 1 8 4 0 3 8
  • C 4 4 5 6
  • D 7 1 1 9 2 2

Q15:

Find, to the nearest second, the measure of the angle between the two straight lines 𝑥 = 2 𝑦 = 3 𝑧 and 5 𝑥 = 5 𝑦 = 𝑧 .

  • A 4 7 3 9 7
  • B 3 1 2 9 2 8
  • C 4 2 2 0 5 3
  • D 5 8 3 0 3 2

Q16:

A straight line 𝐿 passes through the two points 𝐴 ( 6 , 5 , 4 ) and 𝐵 ( 8 , 7 , 8 ) , and a straight line 𝐿 passes through the two points 𝐶 ( 6 , 1 , 5 ) and 𝐷 ( 8 , 3 , 7 ) . Find the measure of the angle between the two lines, giving your answer to two decimal places if necessary.

Q17:

A straight line 𝐿 passes through the two points 𝐴 ( 5 , 1 , 9 ) and 𝐵 ( 7 , 3 , 1 ) , and a straight line 𝐿 passes through the two points 𝐶 ( 1 , 7 , 6 ) and 𝐷 ( 3 , 9 , 2 ) . Find the measure of the angle between the two lines, giving your answer to two decimal places if necessary.

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