Worksheet: Parallel and Perpendicular Lines in Space

In this worksheet, we will practice recognizing parallel and perpendicular lines in space.

Q1:

Determine whether the following is true or false: If the component of a vector in the direction of another vector is zero, then the two are parallel.

  • Afalse
  • Btrue

Q2:

Given that , , and , find the relation between and .

  • A
  • B
  • C
  • D

Q3:

Given that M i j = 2 , L i j = 𝑎 8 , and M L , where i and j are two perpendicular unit vectors, find the value of 𝑎 .

Q4:

Suppose A = 1 , 3 , 2 , B = 𝑘 , 9 , 𝑚 , C = 𝑘 , 𝑚 , 𝑘 + 𝑚 , and A B , find | | C .

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

Q5:

Given the two vectors A i j k = ( 8 7 + ) and B i j k = ( 6 4 5 6 + 8 ) , determine whether these two vectors are parallel, perpendicular, or otherwise.

  • Aotherwise
  • Bperpendicular
  • Cparallel

Q6:

Find the values of 𝑚 and 𝑛 so that vector 2 + 7 + 𝑚 i j k is parallel to vector 6 + 𝑛 2 1 i j k .

  • A 𝑚 = 2 1 , 𝑛 = 7
  • B 𝑚 = 1 . 7 , 𝑛 = 0 . 6
  • C 𝑚 = 2 . 3 , 𝑛 = 6 3
  • D 𝑚 = 7 , 𝑛 = 2 1

Q7:

In the figure, 𝐴 𝐻 is perpendicular to the plane 𝑌 , which contains the points 𝐻 , 𝐵 , 𝐶 , 𝐷 a n d . If 𝐵 𝐷 = 3 6 and 𝐴 𝐷 = 8 5 , find the area of 𝐴 𝐵 𝐷 .

  • A 3,272.5
  • B 1,530
  • C 3,060
  • D 1,386

Q8:

Which of the following vectors is not perpendicular to the line whose direction vector r is 6 , 5 ?

  • A r = 5 , 6
  • B r = 5 , 6
  • C r = 1 0 , 1 2
  • D r = 1 2 , 1 0

Q9:

If the straight line 𝑥 + 8 1 0 = 𝑦 + 8 𝑚 = 𝑧 + 1 0 8 is perpendicular to 𝑥 + 5 4 = 𝑦 + 8 1 0 , and 𝑧 = 8 , find 𝑚 .

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