Worksheet: Equation of a Straight Line in Space: Cartesian and Vector Forms

In this worksheet, we will practice finding the Cartesian and vector forms of the equation of a straight line in space.

Q1:

Find the direction vector of the straight line 𝑥16=𝑦64=𝑧83.

  • A(6,4,3)
  • B(1,6,8)
  • C(6,4,3)
  • D(1,6,8)
  • E(1,6,8)

Q2:

Find the Cartesian form of the equation of the straight line passing through the points (7,3,7) and (3,10,4).

  • A𝑥+74=𝑦+37=𝑧+73
  • B𝑥+73=𝑦+37=𝑧+74
  • C𝑥74=𝑦37=𝑧73
  • D𝑥47=𝑦+73=𝑧37

Q3:

Find the direction vector of the straight line 𝑥36=𝑦+22=𝑧33.

  • A(6,2,3)
  • B(3,2,3)
  • C(6,2,3)
  • D(3,2,3)
  • E(3,2,3)

Q4:

Find the direction vector of the straight line 𝑥64=𝑦36=𝑧17.

  • A(4,6,7)
  • B(6,3,1)
  • C(4,6,7)
  • D(6,3,1)
  • E(6,3,1)

Q5:

Find the direction vector of the straight line 𝑥+15=𝑦42=𝑧75.

  • A(5,2,5)
  • B(1,4,7)
  • C(5,2,5)
  • D(1,4,7)
  • E(1,4,7)

Q6:

Give the Cartesian equation of the line r=3,2,2+𝑡4,2,4.

  • A𝑥+34=𝑦+22=𝑧+24
  • B𝑥43=𝑦22=𝑧42
  • C𝑥+43=𝑦+22=𝑧+42
  • D𝑥34=𝑦+22=𝑧+24

Q7:

Find the Cartesian form of the equation of the straight line passing through the point (4,1,2) and makes equal angles with the coordinates axes.

  • A𝑥4=𝑦1=𝑧2
  • B𝑥+43=𝑦13=𝑧23
  • C𝑥14=𝑦11=𝑧12
  • D𝑥+41=𝑦11=𝑧21

Q8:

Give the Cartesian equation of the line through point (2,5,2) and with direction vector (3,5,4).

  • A𝑥+23=𝑦55=𝑧24
  • B𝑥32=𝑦+55=𝑧+42
  • C𝑥+32=𝑦55=𝑧42
  • D𝑥23=𝑦+55=𝑧+24

Q9:

Give the vector equation of the line through the point (3,7,7) with direction vector 0,5,7.

  • Ar=3,7,7+𝑡0,5,7
  • Br=0,5,7+𝑡3,7,7
  • Cr=3,7,7+𝑡3,7,7
  • Dr=0,5,7+𝑡0,5,7

Q10:

Find the direction vector of the straight line passing through 𝐴1,2,7 and 𝐵4,1,3.

  • Ad=3,1,4
  • Bd=5,3,10
  • Cd=3,1,4
  • Dd=3,1,4

Q11:

Give a direction vector of the line through the origin and the point (6,6,1).

  • Ad=6,6,1
  • Bd=6,6,1
  • Cd=0,0,0
  • Dd=6,6,1

Q12:

Find the direction cosines of the straight line whose direction ratio is 211.

  • A2,22,22, 2,22,22
  • B22,2,2, 22,2,2
  • C63,66,66, 63,66,66
  • D1,12,12, 1,12,12
  • E13,13,13, 13,13,13

Q13:

If the direction cosines of a straight line are 1𝑐,1𝑐,1𝑐, then find the possible values of 𝑐.

  • A2, 2
  • B3, 3
  • C2, 2
  • D3, 3

Q14:

Give the equations for the 𝑧-axis in 3-dimensional space.

  • A𝑥=0, 𝑧=0
  • B𝑥=0, 𝑦=0
  • C𝑧=0
  • D𝑥=1
  • E𝑧=1

Q15:

Give equations for the 𝑥-axis in 3-dimensional space.

  • A𝑦=0, 𝑧=0
  • B𝑧=1
  • C𝑥=1
  • D𝑦=0, 𝑥=0
  • E𝑥=0

Q16:

Give the equations for the 𝑦-axis in 3-dimensional space.

  • A𝑥=0, 𝑦=0
  • B𝑦=0
  • C𝑥=0, 𝑧=0
  • D𝑧=1
  • E𝑦=1

Q17:

Find the vector form of the equation of the straight line passing through the point (2,5,5) and the center of the sphere whose equation is 2𝑥+2𝑦+2𝑧+12𝑥8𝑦+8𝑧=1.

  • Ar=2,5,5+𝑡12,8,8
  • Br=2,5,5+𝑡5,7,3
  • Cr=2,5,5+𝑡5,7,3
  • Dr=2,5,5+𝑡12,8,8

Q18:

Determine, in vector form, the equation of the straight line that passes through the points (5,5,3) and (3,4,4).

  • Ar=2,1,1+𝑡5,5,3
  • Br=5,5,3+𝑡8,9,7
  • Cr=2,1,1+𝑡3,4,4
  • Dr=5,5,3+𝑡2,1,1

Q19:

The points 𝐴(8,9,2), 𝐵(0,7,6), and 𝐶(8,1,4) form a triangle. Determine, in vector form, the equation of the median drawn from 𝐶.

  • Ar=8,2,8+𝑡8,1,4
  • Br=8,1,4+𝑡8,2,8
  • Cr=8,1,4+𝑡4,7,6
  • Dr=4,7,6+𝑡8,1,4

Q20:

Find the vector form of the equation of the straight line 4𝑥39=7𝑦82=7+6𝑧4.

  • Ar=43,78,67+𝑡49,72,32
  • Br=34,87,76+𝑡94,27,23
  • Cr=34,87,76+𝑡94,27,23
  • Dr=94,27,23+𝑡34,87,76

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