Lesson Worksheet: Parallel, Perpendicular, and Intersecting Lines in Space Mathematics

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:

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

Q2:

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

Q3:

Find the vector form of the equation of the straight line passing through 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

Q4:

The lines 𝑥=3𝑡2, 𝑦=3𝑡+2, 𝑧=9𝑡2 and 𝑥=𝑎𝑡2, 𝑦=𝑡+1, 𝑧=𝑏𝑡2 are parallel. What is 𝑎+𝑏?

Q5:

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

Q6:

If the two straight lines 𝑥+38=𝑦44𝑛=𝑧+110 and 𝑥+54𝑛=𝑦+104=𝑧35 are perpendicular, find 𝑛.

  • A2524
  • B2425
  • C2425
  • D2524

Q7:

Consider the two lines 𝑥=4+2𝑡, 𝑦=6+𝑡, 𝑧=22𝑡 and r=6,7,0+𝑡5,4,7. Determine whether they are parallel or perpendicular.

  • AParallel
  • BPerpendicular

Q8:

Consider the two lines 𝐿: r=3,4,1+𝑡1,1,1 and 𝐿: r=2,0,5+𝑡2,3,0. Choose the correct statement about them.

  • A𝐿 and 𝐿 are parallel but not coincident.
  • B𝐿 and 𝐿 are coincident lines.
  • C𝐿 and 𝐿 are skew lines.
  • D𝐿 and 𝐿 intersect at a point

Q9:

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
  • B𝑥2=𝑦4=𝑧3
  • C𝑥9=𝑦3=𝑧4
  • D𝑥3=𝑦9=𝑧4

Q10:

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

This lesson includes 18 additional questions and 207 additional question variations for subscribers.

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