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.

  • A8356′57′′∘
  • B7239′14′′∘
  • C622′46′′∘
  • D752′12′′∘

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).

  • A8756′48′′∘
  • B6336′3′′∘
  • C2623′57′′∘
  • D889′39′′∘

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.

  • A2953′9′′∘
  • B7616′39′′∘
  • C606′51′′∘
  • D1343′21′′∘

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.

  • A8136′37′′∘
  • B999′30′′∘
  • C4731′25′′∘
  • D8050′29′′∘

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.

  • A6028′26′′∘
  • B5426′18′′∘
  • C12533′41′′∘
  • D7044′29′′∘

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.

  • A2014′∘
  • B402′33′′∘
  • C3153′48′′∘
  • D13957′26′′∘

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√2121
  • B2√2121
  • C4√2121
  • D√2121

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.

  • A8227′18′′∘
  • B6811′55′′∘
  • C2311′55′′∘
  • D7114′18′′∘

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.

  • A6225′30′′∘
  • B647′24′′∘
  • C7124′50′′∘
  • D5057′15′′∘

Q14:

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

  • A445′6′′∘
  • B7119′22′′∘
  • C4554′54′′∘
  • D1840′38′′∘

Q15:

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

  • A4220′53′′∘
  • B5830′32′′∘
  • C4739′7′′∘
  • D3129′28′′∘

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)
  • D1√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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