Worksheet: Angle between Two Straight Lines in Space

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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=𝑦−2−4=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 𝐿∶𝑥=5−8𝑡, 𝑦=−3−4𝑡, 𝑧=5+6𝑡 and 𝐿∶𝑥−53=𝑦+5−6=𝑧−2−2, 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 313√2,911√2,−37√2 and −1013√2,−813√2,98√2. 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+𝑡−27,−43,95 and −6𝑥−27=4𝑦−3−6=3−8𝑧−5.

  • 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 𝐿∶𝑥=0, 𝑦=𝑧 and 𝐿∶𝑦=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=𝑦−1−3=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=𝑦−14=𝑧−2−1 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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