Worksheet: Angle between Two Straight Lines in Three Dimensions

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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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