Worksheet: Direction Cosines of a Straight Line

In this worksheet, we will practice finding the direction cosine of a vector or a straight line from the direction ratios or that of a normal to a plane given the plane equation.

Q1:

Which of the following gives the direction cosines of a straight line?

  • A ( 1 , 1 , 1 )
  • B ( 1 , 1 , 1 )
  • C ( 3 , 3 , 3 )
  • D 1 3 , 1 3 , 1 3
  • E ( 1 . 5 , 1 . 5 , 1 . 5 )

Q2:

Find the direction cosines of the straight line whose direction ratio is 2 1 1 .

  • A 2 , 2 2 , 2 2 , 2 , 2 2 , 2 2
  • B 1 3 , 1 3 , 1 3 , 1 3 , 1 3 , 1 3
  • C 2 2 , 2 , 2 , 2 2 , 2 , 2
  • D 6 3 , 6 6 , 6 6 , 6 3 , 6 6 , 6 6
  • E 1 , 1 2 , 1 2 , 1 , 1 2 , 1 2

Q3:

The given figure represents a vector 𝑂 𝐴 whose norm is 5 units. Find the measure of the direction angles of 𝑂 𝐴 rounded to one decimal place.

  • A 𝜃 = 5 0 . 6 , 𝜃 = 2 4 . 2 , 𝜃 = 2 9 . 0
  • B 𝜃 = 3 9 . 4 , 𝜃 = 6 5 . 8 , 𝜃 = 6 1 . 0
  • C 𝜃 = 3 7 . 7 , 𝜃 = 2 2 . 3 , 𝜃 = 2 2 . 3
  • D 𝜃 = 6 5 . 8 , 𝜃 = 3 9 . 4 , 𝜃 = 6 1 . 0

Q4:

Suppose that 𝐴 = 6 and 𝐴 has direction cosines 2 3 , 2 3 , and 1 3 . Determine 𝐴 × 𝐵 , where 𝐵 = ( 8 , 0 , 3 ) .

  • A 3 2 𝑖 1 2 𝑗 + 4 𝑘
  • B 1 2 𝑖 4 𝑗 3 2 𝑘
  • C 1 2 𝑖 + 2 8 𝑗 + 3 2 𝑘
  • D 1 2 𝑖 + 4 𝑗 3 2 𝑘

Q5:

Find the direction cosines of the normal to the plane 4 𝑥 + 8 𝑦 3 𝑧 = 2 8 .

  • A 1 4 , 1 2 , 3 1 6
  • B 4 1 5 1 5 , 8 1 5 1 5 , 1 5 5
  • C 4 8 9 , 8 8 9 , 3 8 9
  • D 4 8 9 8 9 , 8 8 9 8 9 , 3 8 9 8 9

Q6:

If the direction cosines of a straight line are 1 𝑐 , 1 𝑐 , 1 𝑐 , then find the possible values of 𝑐 .

  • A3, 3
  • B 2 , 2
  • C2, 2
  • D 3 , 3

Q7:

Given that 𝑙 , 𝑚 , and 𝑛 are the direction cosines of a straight line, find the value of 𝑙 + 𝑚 + 𝑛 2 2 2 .

Q8:

If the direction angles of a straight line are 𝑇 𝑥 , 𝑇 𝑦 , and 𝑇 𝑧 , then find the value of c o s c o s c o s 2 𝑇 + 2 𝑇 + 2 𝑇 𝑥 𝑦 𝑧 .

Q9:

Find the direction cosines of the straight line whose direction ratio is 4 5 2 .

  • A 4 7 7 , 5 7 7 , 2 7 7 , 4 7 7 , 5 7 7 , 2 7 7
  • B 4 4 5 , 4 4 5 , 4 4 5 , 4 4 5 , 4 4 5 , 4 4 5
  • C 4 7 , 5 7 , 2 7 , 4 7 , 5 7 , 2 7
  • D 4 5 1 5 , 5 3 , 2 5 1 5 , 4 5 1 5 , 5 3 , 2 5 1 5
  • E 4 7 , 5 7 , 2 7 , 4 7 , 5 7 , 2 7

Q10:

Find the direction cosines of the normal to the plane 8 𝑥 + 5 𝑦 2 𝑧 = 6 .

  • A 1 8 , 5 6 4 , 1 3 2
  • B 8 1 5 1 5 , 1 5 3 , 2 1 5 1 5
  • C 8 9 3 , 5 9 3 , 2 9 3
  • D 8 9 3 9 3 , 5 9 3 9 3 , 2 9 3 9 3

Q11:

Find the direction cosines of the normal to the plane 6 𝑥 + 7 𝑦 + 3 𝑧 = 6 .

  • A 1 6 , 7 3 6 , 1 1 2
  • B 3 2 , 7 4 , 3 4
  • C 3 4 7 , 7 9 4 , 3 9 4
  • D 3 9 4 4 7 , 7 9 4 9 4 , 3 9 4 9 4

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