Worksheet: Angle between Two Straight Lines in Space

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

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+12=𝑦24=𝑧+25 and the positive direction of the 𝑥-axis.

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

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

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 2𝑥=4𝑦=3𝑧 and 4𝑥=5𝑦=2𝑧.

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

Q6:

Find the measure of the angle between the two straight lines 𝐿𝑥=58𝑡, 𝑦=34𝑡, 𝑧=5+6𝑡 and 𝐿𝑥53=𝑦+56=𝑧22, and round it to the nearest second.

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

Q7:

Find the measure of the angle between the two straight lines whose direction cosines are 3132,9112,372 and 10132,8132,982. Give your answer to the nearest second.

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

Q8:

Find the measure of the angle between the two straight lines 𝑟=27,23,1+𝑡27,43,95 and 6𝑥27=4𝑦36=38𝑧5.

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

Q9:

If 𝐿𝑥=0, 𝑦=𝑧 and 𝐿𝑦=0, 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 cos𝜃.

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

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

Q11:

If a straight line makes direction angles of measures 60 with the 𝑦-axis and 60 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+12=𝑦13=𝑧+44 and the positive direction of the 𝑥-axis.

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

Q13:

Find, to the nearest second, the measure of the angle between the straight line 𝑥+52=𝑦14=𝑧21 and the positive direction of the 𝑥-axis.

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

Q14:

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

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

Q15:

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

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

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.

Q18:

Which of the following gives the direction cosines of a straight line?

  • A ( 1 . 5 , 1 . 5 , 1 . 5 )
  • B ( 3 , 3 , 3 )
  • C ( 1 , 1 , 1 )
  • D 1 3 , 1 3 , 1 3
  • E ( 1 , 1 , 1 )

Q19:

Given that 𝑙, 𝑚, and 𝑛 are the direction cosines of a straight line, find the value of 𝑙+𝑚+𝑛.

Q20:

If the direction angles of a straight line are 𝑇, 𝑇, and 𝑇, then find the value of coscoscos2𝑇+2𝑇+2𝑇.

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