Worksheet: Equations of Parallel and Perpendicular Lines in Space

In this worksheet, we will practice finding the equation of a straight line that is parallel or perpendicular to another one in space and finding the intersection point between two lines.

Q1:

The straight line that passes through the point 𝐴5,4 is perpendicular to the vector u=5,2. Find the Cartesian equation of the line.

  • A5𝑥2𝑦+17=0
  • B5𝑥2𝑦=0
  • C4𝑥5𝑦=0
  • D4𝑥5𝑦17=0

Q2:

What is the equation of the line parallel to line 𝑥+99=𝑦+41=𝑧47 and passing through (2,5,6)?

  • A𝑥92=𝑦45=𝑧+46
  • B𝑥+29=𝑦+51=𝑧67
  • C𝑥29=𝑦51=𝑧+67
  • D𝑥+92=𝑦+45=𝑧46

Q3:

Find the Cartesian form of the equation of the straight line passing through the origin and the intersection point of the two straight lines 𝐿=1,1,2+𝑡1,4,3r and 𝐿𝑥=3, 𝑦54=𝑧31.

  • A𝑥9=𝑦3=𝑧4
  • B𝑥3=𝑦9=𝑧4
  • C𝑥9=𝑦3=𝑧4
  • D𝑥3=𝑦9=𝑧4

Q4:

Find the vector form of the equation of the straight line passing throught the point 𝐴(2,5,5) and parallel to the straight line passing through the two points 𝐵(3,2,6) and 𝐶(5,0,9).

  • Ar=2,5,5+𝑡8,2,3
  • Br=2,5,5+𝑡3,2,6
  • Cr=2,5,5+𝑡5,0,9
  • Dr=8,2,3+𝑡2,5,5

Q5:

Find the vector form of the equation of the straight line passing through the point (5,1,4) and the intersection point of the two straight lines 𝑥+22=𝑦+52=𝑧+31 and 𝑥+13=𝑦12=𝑧+32.

  • Ar=5,1,4+𝑡7,2,9
  • Br=5,1,4+𝑡7,2,9
  • Cr=5,1,4+𝑡7,2,9
  • Dr=5,1,4+𝑡7,2,9

Q6:

Determine the vector form of the equation of the straight line passing through the point (1,5,4) and parallel to the vector 3,5,1.

  • Ar=1,5,4+𝑡2,10,3
  • Br=3,5,1+𝑡1,5,4
  • Cr=1,5,4+𝑡3,5,1
  • Dr=3,5,1+𝑡2,10,3

Q7:

Given that 𝐿𝑥+97=𝑦37=𝑧+86 is perpendicular to 𝐿𝑥29=𝑦10𝑘=𝑧+3𝑚, what is 7𝑘+6𝑚?

Q8:

Given that the vector A=2,𝑘,6 is parallel to the line 𝑥63=𝑦56=𝑧+49, find 𝑘.

Q9:

Find the parametric equations of the straight line that passes through the point 𝐴(1,4,1) and that is parallel to the bisector of the second quadrant of the plane 𝑦𝑧.

  • A𝑥=1+12𝑡, 𝑦=4+12𝑡, 𝑧=1+12𝑡
  • B𝑥=1+𝑡, 𝑦=4+𝑡, 𝑧=1+𝑡
  • C𝑥=1+𝑡, 𝑦=4+12𝑡, 𝑧=1+12𝑡
  • D𝑥=1, 𝑦=4𝑡, 𝑧=1+𝑡

Q10:

Suppose that lines r=5,3,4+𝑡3,1,𝑔 and 𝑥5=𝑦44=𝑧24 are parallel, what are 𝑔 and ?

  • A𝑔=12, =1
  • B𝑔=1, =12
  • C𝑔=12, =1
  • D𝑔=1, =12
  • E𝑔=1, =12

Q11:

Given that the lines 𝑥83=𝑦+45=𝑧+62 and 𝑥105=𝑦+79=𝑧3𝑚 are perpendicular, what is 𝑚?

Q12:

Find the equation of the line through the origin that intersects the line r=1,2,3+𝑡3,5,1 orthogonally.

  • Ar=𝑡19,31,13
  • Br=𝑡13,24,19
  • Cr=𝑡23,39,1
  • Dr=𝑡1,4,23

Q13:

For what value of 𝑎 do the lines 𝑥5=𝑦21=𝑧2 and 𝑥1𝑎=𝑦+24=𝑧+14 intersect?

Q14:

For what values of 𝑘 is the line 𝐿𝑥82=𝑦105=𝑧+13 parallel to the line 𝐿𝑥210=𝑦2𝑘+2=𝑧615?

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