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 ๐‘ฅ ๐‘ง and makes an angle of 6 0 โˆ˜ with the positive ๐‘ง -axis.

  • A ๏€ฟ 0 , โˆš 3 2 , 1 2 ๏‹
  • B ๏€ฟ โˆš 3 2 , 1 2 , 0 ๏‹
  • C ๏€ฟ 1 2 , 0 , โˆš 3 2 ๏‹
  • D ๏€ฟ 0 , 1 2 , โˆš 3 2 ๏‹
  • E ๏€ฟ โˆš 3 2 , 0 , 1 2 ๏‹

Q2:

Find the vector A of norm 61 and direction cosines ๏€ฟ 1 2 , โˆ’ 1 2 , โˆš 2 2 ๏‹ .

  • A A = ๏“’ 1 2 , โˆ’ 1 2 , โˆš 2 2 ๏““
  • B A = ๏“’ 6 1 โˆš 3 2 , 6 1 โˆš 3 2 , 6 1 โˆš 2 2 ๏““
  • C A = ๏‡ฒ 6 1 โˆš 3 , โˆ’ 6 1 โˆš 3 , 6 1 ๏‡ถ
  • D A = ๏“’ โˆš 3 2 , โˆš 3 2 , โˆš 2 2 ๏““
  • E A = ๏“’ 6 1 2 , โˆ’ 6 1 2 , 6 1 โˆš 2 2 ๏““

Q3:

The direction angles of A are 9 0 โˆ˜ , 9 7 โˆ˜ , and 1 6 5 โˆ˜ . Which of the following planes contains A ?

  • A ๐‘ฅ ๐‘ง
  • B ๐‘ฅ ๐‘ฆ
  • C ๐‘ฆ ๐‘ง

Q4:

Find vector A whose norm is 41 and whose direction angles are ( 1 3 5 , 1 2 0 , 6 0 ) โˆ˜ โˆ˜ โˆ˜ .

  • A A = ๏“’ โˆ’ โˆš 2 2 , โˆ’ 1 2 , 1 2 ๏““
  • B A = ๏“’ 4 1 โˆš 2 2 , 4 1 โˆš 3 2 , 4 1 โˆš 3 2 ๏““
  • C A = ๏‡ฒ โˆ’ 4 1 , โˆ’ 4 1 โˆš 3 , 4 1 โˆš 3 ๏‡ถ
  • D A = ๏“’ โˆ’ 4 1 โˆš 2 2 , โˆ’ 4 1 2 , 4 1 2 ๏““

Q5:

Determine the angle between the vectors and in .

  • A
  • B
  • C0
  • D

Q6:

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 = ยฑ 3 1 โˆš 3 ( + + )
  • C A i j k = ยฑ 3 1 โˆš 3 3 ( + + )

Q7:

Find the direction angles of the vector .

  • A
  • B
  • C
  • D
  • E

Q8:

If A i j k = 9 โˆ’ + 6 , find the measure of the angle that A makes with the positive direction of the ๐‘ฅ -axis approximated to the nearest second.

  • A 7 3 9 โ€ฒ 3 5 โ€ฒ โ€ฒ โˆ˜
  • B 3 2 4 4 โ€ฒ 7 โ€ฒ โ€ฒ โˆ˜
  • C 5 5 5 6 โ€ฒ 4 8 โ€ฒ โ€ฒ โˆ˜
  • D 3 4 3 โ€ฒ 1 2 โ€ฒ โ€ฒ โˆ˜

Q9:

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

  • A โŸจ 1 , 1 , 1 โŸฉ
  • B ๏‡ณ 3 , 1 5 2 , 1 5 8 ๏‡ท
  • C โŸจ 5 , 2 , 8 โŸฉ
  • D ๏“’ 5 โˆš 9 3 , 2 โˆš 9 3 , 8 โˆš 9 3 ๏““

Q10:

Suppose that 3 1 โˆ˜ , 6 5 โˆ˜ and ๐œƒ are the direction angles of a vector. Which of the following, to the nearest hundredth, is ๐œƒ ?

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

Q11:

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

  • A ๐œƒ = 3 2 . 4 ๏— โˆ˜ , ๐œƒ = 6 9 . 2 ๏˜ โˆ˜ , ๐œƒ = 6 6 . 4 ๏™ โˆ˜
  • B ๐œƒ = 2 0 . 8 ๏— โˆ˜ , ๐œƒ = 5 7 . 6 ๏˜ โˆ˜ , ๐œƒ = 2 3 . 6 ๏™ โˆ˜
  • C ๐œƒ = 1 9 . 6 ๏— โˆ˜ , ๐œƒ = 4 0 . 2 ๏˜ โˆ˜ , ๐œƒ = 2 1 . 8 ๏™ โˆ˜
  • D ๐œƒ = 6 9 . 2 ๏— โˆ˜ , ๐œƒ = 3 2 . 4 ๏˜ โˆ˜ , ๐œƒ = 6 6 . 4 ๏™ โˆ˜

Q12:

Find, to the nearest second, the measure of the angle that vector A = โŸจ โˆ’ 9 , 4 , 8 โŸฉ makes with the ๐‘ฅ -axis.

  • A 7 1 โˆ˜ 3 7 โ€ฒ 2 8 โ€ฒ โ€ฒ
  • B 5 0 โˆ˜ 5 4 โ€ฒ 5 0 โ€ฒ โ€ฒ
  • C 1 3 5 โˆ˜ 1 0 โ€ฒ 4 1 โ€ฒ โ€ฒ

Q13:

Find the vector A of norm 59 and direction cosines ๏€ฟ โˆ’ 1 2 , โˆš 2 2 , โˆ’ 1 2 ๏‹ .

  • A A = ๏“’ โˆ’ 1 2 , โˆš 2 2 , โˆ’ 1 2 ๏““
  • B A = ๏“’ 5 9 โˆš 3 2 , 5 9 โˆš 2 2 , 5 9 โˆš 3 2 ๏““
  • C A = ๏‡ฒ โˆ’ 5 9 โˆš 3 , 5 9 , โˆ’ 5 9 โˆš 3 ๏‡ถ
  • D A = ๏“’ โˆš 3 2 , โˆš 2 2 , โˆš 3 2 ๏““
  • E A = ๏“’ โˆ’ 5 9 2 , 5 9 โˆš 2 2 , โˆ’ 5 9 2 ๏““

Q14:

The direction angles of A are 6 7 โˆ˜ , 4 9 โˆ˜ , and 9 0 โˆ˜ . Which of the following planes contains A ?

  • A ๐‘ฅ ๐‘ง
  • B ๐‘ฆ ๐‘ง
  • C ๐‘ฅ ๐‘ฆ

Q15:

The direction angles of A are 1 0 3 โˆ˜ , 9 0 โˆ˜ , and 8 2 โˆ˜ . Which of the following planes contains A ?

  • A ๐‘ฆ ๐‘ง
  • B ๐‘ฅ ๐‘ฆ
  • C ๐‘ฅ ๐‘ง

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