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Lesson: Angle between Two Straight Lines in Three Dimensions

Worksheet • 17 Questions

Q1:

A straight line 𝐿 1 passes through the two points 𝐴 ( βˆ’ 2 , 2 , βˆ’ 3 ) and 𝐡 ( βˆ’ 6 , βˆ’ 4 , βˆ’ 5 ) , and a straight line 𝐿 2 passes through the two points 𝐢 ( 1 , 4 , 1 ) and 𝐷 ( βˆ’ 9 , βˆ’ 6 , βˆ’ 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 π‘₯ + 1 2 = 𝑦 βˆ’ 2 βˆ’ 4 = 𝑧 + 2 5 and the positive direction of the π‘₯ -axis.

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

Q3:

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

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

Q4:

Find the measure of the angle between the straight line π‘₯ = 1 , 𝑦 = 2 and the straight line 𝑦 = βˆ’ 1 , 𝑧 = 0 .

Q5:

Find, to the nearest second, the measure of the angle between the two straight lines βˆ’ 2 π‘₯ = 4 𝑦 = βˆ’ 3 𝑧 and βˆ’ 4 π‘₯ = βˆ’ 5 𝑦 = 2 𝑧 .

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

Q6:

Find the measure of the angle between the two straight lines 𝐿 ∢ π‘₯ = 5 βˆ’ 8 𝑑  , 𝑦 = βˆ’ 3 βˆ’ 4 𝑑 , 𝑧 = 5 + 6 𝑑 and 𝐿 ∢ π‘₯ βˆ’ 5 3 = 𝑦 + 5 βˆ’ 6 = 𝑧 βˆ’ 2 βˆ’ 2  , and round it to the nearest second.

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

Q7:

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

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

Q8:

Find the measure of the angle between the two straight lines ⃑ π‘Ÿ = ο€Ό 2 7 , βˆ’ 2 3 , βˆ’ 1  + 𝑑 ο€Ό βˆ’ 2 7 , βˆ’ 4 3 , 9 5  1 1 and βˆ’ 6 π‘₯ βˆ’ 2 7 = 4 𝑦 βˆ’ 3 βˆ’ 6 = 3 βˆ’ 8 𝑧 βˆ’ 5 .

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

Q9:

If 𝐿 ∢ π‘₯ = 0 1 , 𝑦 = 𝑧 and 𝐿 ∢ 𝑦 = 0 2 , π‘₯ = 𝑧 , then find the value of πœƒ .

Q10:

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

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

  • A 2 √ 2 1 2 1
  • B 4 √ 2 1 2 1
  • C βˆ’ 4 √ 2 1 2 1
  • D √ 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 π‘₯ -axis.

Q12:

Find, to the nearest second, the measure of the angle between the straight line π‘₯ + 1 2 = 𝑦 βˆ’ 1 βˆ’ 3 = 𝑧 + 4 4 and the positive direction of the π‘₯ -axis.

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

Q13:

Find, to the nearest second, the measure of the angle between the straight line π‘₯ + 1 √ 2 = 𝑦 βˆ’ √ 2 βˆ’ 3 = 𝑧 + 1 1 and the positive direction of the 𝑧 -axis.

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

Q14:

Find, to the nearest second, the measure of the angle between the two straight lines 3 π‘₯ = βˆ’ 2 𝑦 = βˆ’ 𝑧 and 5 π‘₯ = 𝑦 = 5 𝑧 .

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

Q15:

Find, to the nearest second, the measure of the angle between the two straight lines βˆ’ 5 π‘₯ = 𝑦 = 3 𝑧 and βˆ’ 3 π‘₯ = βˆ’ 3 𝑦 = 4 𝑧 .

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

Q16:

A straight line 𝐿 1 passes through the two points 𝐴 ( βˆ’ 6 , 5 , βˆ’ 4 ) and 𝐡 ( βˆ’ 8 , βˆ’ 7 , βˆ’ 8 ) , and a straight line 𝐿 2 passes through the two points 𝐢 ( βˆ’ 6 , 1 , βˆ’ 5 ) and 𝐷 ( βˆ’ 8 , βˆ’ 3 , βˆ’ 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 1 passes through the two points 𝐴 ( 5 , 1 , 9 ) and 𝐡 ( βˆ’ 7 , βˆ’ 3 , βˆ’ 1 ) , and a straight line 𝐿 2 passes through the two points 𝐢 ( 1 , 7 , 6 ) and 𝐷 ( βˆ’ 3 , βˆ’ 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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