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.

  • A0,12,32
  • B0,32,12
  • C32,0,12
  • D32,12,0
  • E12,0,32

Q2:

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

  • AA=6132,6132,6122
  • BA=12,12,22
  • CA=613,613,61
  • DA=612,612,6122
  • EA=32,32,22

Q3:

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

  • A𝑥𝑦
  • B𝑦𝑧
  • C𝑥𝑧

Q4:

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

  • AA=41,413,413
  • BA=4122,4132,4132
  • CA=4122,412,412
  • DA=22,12,12

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.

  • AAijk=10.33(++)
  • BAijk=±313(++)
  • CAijk=±3133(++)

Q6:

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

  • A(60,45,60)
  • B(35,27,27)
  • C(45,60,60)
  • D(27,35,27)
  • E(30,45,30)

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.

  • A73935
  • B34312
  • C555648
  • D32447

Q8:

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

  • A593,293,893
  • B3,152,158
  • C1,1,1
  • D5,2,8

Q9:

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

  • A72.88
  • B85.03
  • C264.00
  • D84.00

Q10:

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

  • A𝜃=69.2, 𝜃=32.4, 𝜃=66.4
  • B𝜃=19.6, 𝜃=40.2, 𝜃=21.8
  • C𝜃=20.8, 𝜃=57.6, 𝜃=23.6
  • D𝜃=32.4, 𝜃=69.2, 𝜃=66.4

Q11:

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

  • A505450
  • B1351041
  • C713728

Q12:

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

  • AA=5932,5922,5932
  • BA=12,22,12
  • CA=593,59,593
  • DA=592,5922,592
  • EA=32,22,32

Q13:

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

  • A𝑦𝑧
  • B𝑥𝑦
  • C𝑥𝑧

Q14:

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

  • A𝑥𝑦
  • B𝑦𝑧
  • C𝑥𝑧

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.

  • A𝜃=50.6, 𝜃=24.2, 𝜃=29.0
  • B𝜃=37.7, 𝜃=22.3, 𝜃=22.3
  • C𝜃=39.4, 𝜃=65.8, 𝜃=61.0
  • D𝜃=65.8, 𝜃=39.4, 𝜃=61.0

Q16:

The vector 𝑂𝐴 has length 10. Write this vector using i, j, and k, with components accurate to two decimal places.

  • AAijk=8.72+3.35+3.58
  • BAijk=1.28+3.35+9.34
  • CAijk=3.58+9.34+9.34
  • DAijk=3.35+8.72+3.58

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