Worksheet: Direction Angles and Direction Cosines

In this worksheet, we will practice finding direction angles and direction cosines for a given vector in space.

Q1:

Find the direction cosines of the vector that lies in the positive coordinate plane 𝑥𝑧 and makes an angle of 60 with the positive 𝑧-axis.

  • A 0 , 1 2 , 3 2
  • B 0 , 3 2 , 1 2
  • C 3 2 , 0 , 1 2
  • D 3 2 , 1 2 , 0
  • E 1 2 , 0 , 3 2

Q2:

Find the vector A of norm 61 and direction cosines 12,12,22.

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

Q3:

The direction angles of A are 90, 97, and 165. Which of the following planes contains A?

  • A 𝑥 𝑦
  • B 𝑦 𝑧
  • C 𝑥 𝑧

Q4:

Find vector A whose norm is 41 and whose direction angles are (135,120,60).

  • A A = 4 1 , 4 1 3 , 4 1 3
  • B A = 4 1 2 2 , 4 1 3 2 , 4 1 3 2
  • C A = 4 1 2 2 , 4 1 2 , 4 1 2
  • D A = 2 2 , 1 2 , 1 2

Q5:

Find the algebraic form of a vector A if its norm is 31, given that it makes equal angles with the positive directions of the Cartesian axes.

  • A A i j k = 1 0 . 3 3 ( + + )
  • B A i j k = ± 3 1 3 ( + + )
  • C A i j k = ± 3 1 3 3 ( + + )

Q6:

Find the direction angles of the vector 212,2122,212.

  • A ( 6 0 , 4 5 , 6 0 )
  • B ( 3 5 , 2 7 , 2 7 )
  • C ( 4 5 , 6 0 , 6 0 )
  • D ( 2 7 , 3 5 , 2 7 )
  • E ( 3 0 , 4 5 , 3 0 )

Q7:

If Aijk=9+6, find the measure of the angle that A makes with the positive direction of the 𝑥-axis approximated to the nearest second.

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

Q8:

Find the direction cosines of the vector A=5,2,8.

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

Q9:

Suppose that 31, 65 and 𝜃 are the direction angles of a vector. Which of the following, to the nearest hundredth, is 𝜃?

  • A 7 2 . 8 8
  • B 8 5 . 0 3
  • C 2 6 4 . 0 0
  • D 8 4 . 0 0

Q10:

Find the measure of the direction angles of the vector F, represented by the given figure, corrected to one decimal place.

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

Q11:

Find, to the nearest second, the measure of the angle that vector A=9,4,8 makes with the 𝑥-axis.

  • A 5 0 5 4 5 0
  • B 1 3 5 1 0 4 1
  • C 7 1 3 7 2 8

Q12:

Find the vector A of norm 59 and direction cosines 12,22,12.

  • A A = 5 9 3 , 5 9 , 5 9 3
  • B A = 5 9 2 , 5 9 2 2 , 5 9 2
  • C A = 1 2 , 2 2 , 1 2
  • D A = 5 9 3 2 , 5 9 2 2 , 5 9 3 2
  • E A = 3 2 , 2 2 , 3 2

Q13:

The direction angles of A are 67, 49, and 90. Which of the following planes contains A?

  • A 𝑦 𝑧
  • B 𝑥 𝑦
  • C 𝑥 𝑧

Q14:

The direction angles of A are 90, 143, and 53. Which of the following planes contains A?

  • A 𝑥 𝑦
  • B 𝑦 𝑧
  • C 𝑥 𝑧

Q15:

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

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

Q16:

Suppose that ||=6A and A has direction cosines 23, 23, and 13. Determine AB×, where B=8,0,3.

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

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