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.

  • A835657
  • B723914
  • C62246
  • D75212

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).

  • A875648
  • B63363
  • C262357
  • D88939

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.

  • A29539
  • B761639
  • C60651
  • D134321

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.

  • A813637
  • B99930
  • C473125
  • D805029

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.

  • A602826
  • B542618
  • C1253341
  • D704429

Q8:

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

  • A2014
  • B40233
  • C315348
  • D1395726

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.

  • A42121
  • B22121
  • C42121
  • D2121

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.

  • A822718
  • B681155
  • C231155
  • D711418

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.

  • A622530
  • B64724
  • C712450
  • D505715

Q14:

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

  • A4456
  • B711922
  • C455454
  • D184038

Q15:

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

  • A422053
  • B583032
  • C47397
  • D312928

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)
  • D13,13,13
  • 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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