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).

  • A 𝑥 + 7 4 = 𝑦 + 3 7 = 𝑧 + 7 3
  • B 𝑥 + 7 3 = 𝑦 + 3 7 = 𝑧 + 7 4
  • C 𝑥 7 4 = 𝑦 3 7 = 𝑧 7 3
  • D 𝑥 4 7 = 𝑦 + 7 3 = 𝑧 3 7

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.

  • A 𝑥 + 3 4 = 𝑦 + 2 2 = 𝑧 + 2 4
  • B 𝑥 4 3 = 𝑦 2 2 = 𝑧 4 2
  • C 𝑥 + 4 3 = 𝑦 + 2 2 = 𝑧 + 4 2
  • D 𝑥 3 4 = 𝑦 + 2 2 = 𝑧 + 2 4

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.

  • A 𝑥 4 = 𝑦 1 = 𝑧 2
  • B 𝑥 + 4 3 = 𝑦 1 3 = 𝑧 2 3
  • C 𝑥 1 4 = 𝑦 1 1 = 𝑧 1 2
  • D 𝑥 + 4 1 = 𝑦 1 1 = 𝑧 2 1

Q8:

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

  • A 𝑥 + 2 3 = 𝑦 5 5 = 𝑧 2 4
  • B 𝑥 3 2 = 𝑦 + 5 5 = 𝑧 + 4 2
  • C 𝑥 + 3 2 = 𝑦 5 5 = 𝑧 4 2
  • D 𝑥 2 3 = 𝑦 + 5 5 = 𝑧 + 2 4

Q9:

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

  • A r = 3 , 7 , 7 + 𝑡 0 , 5 , 7
  • B r = 0 , 5 , 7 + 𝑡 3 , 7 , 7
  • C r = 3 , 7 , 7 + 𝑡 3 , 7 , 7
  • D r = 0 , 5 , 7 + 𝑡 0 , 5 , 7

Q10:

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

  • A d = 3 , 1 , 4
  • B d = 5 , 3 , 1 0
  • C d = 3 , 1 , 4
  • D d = 3 , 1 , 4

Q11:

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

  • A d = 6 , 6 , 1
  • B d = 6 , 6 , 1
  • C d = 0 , 0 , 0
  • D d = 6 , 6 , 1

Q12:

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

  • A 2 , 2 2 , 2 2 , 2 , 2 2 , 2 2
  • B 2 2 , 2 , 2 , 2 2 , 2 , 2
  • C 6 3 , 6 6 , 6 6 , 6 3 , 6 6 , 6 6
  • D 1 , 1 2 , 1 2 , 1 , 1 2 , 1 2
  • E 1 3 , 1 3 , 1 3 , 1 3 , 1 3 , 1 3

Q13:

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

  • A 2 , 2
  • B3, 3
  • C2, 2
  • D 3 , 3

Q14:

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

  • A 𝑥 = 0 , 𝑧 = 0
  • B 𝑥 = 0 , 𝑦 = 0
  • C 𝑧 = 0
  • D 𝑥 = 1
  • E 𝑧 = 1

Q15:

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

  • A 𝑦 = 0 , 𝑧 = 0
  • B 𝑧 = 1
  • C 𝑥 = 1
  • D 𝑦 = 0 , 𝑥 = 0
  • E 𝑥 = 0

Q16:

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

  • A 𝑥 = 0 , 𝑦 = 0
  • B 𝑦 = 0
  • C 𝑥 = 0 , 𝑧 = 0
  • D 𝑧 = 1
  • E 𝑦 = 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.

  • A r = 2 , 5 , 5 + 𝑡 1 2 , 8 , 8
  • B r = 2 , 5 , 5 + 𝑡 5 , 7 , 3
  • C r = 2 , 5 , 5 + 𝑡 5 , 7 , 3
  • D r = 2 , 5 , 5 + 𝑡 1 2 , 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).

  • A r = 2 , 1 , 1 + 𝑡 5 , 5 , 3
  • B r = 5 , 5 , 3 + 𝑡 8 , 9 , 7
  • C r = 2 , 1 , 1 + 𝑡 3 , 4 , 4
  • D r = 5 , 5 , 3 + 𝑡 2 , 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.

  • A r = 8 , 2 , 8 + 𝑡 8 , 1 , 4
  • B r = 8 , 1 , 4 + 𝑡 8 , 2 , 8
  • C r = 8 , 1 , 4 + 𝑡 4 , 7 , 6
  • D r = 4 , 7 , 6 + 𝑡 8 , 1 , 4

Q20:

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

  • A r = 4 3 , 7 8 , 6 7 + 𝑡 4 9 , 7 2 , 3 2
  • B r = 3 4 , 8 7 , 7 6 + 𝑡 9 4 , 2 7 , 2 3
  • C r = 3 4 , 8 7 , 7 6 + 𝑡 9 4 , 2 7 , 2 3
  • D r = 9 4 , 2 7 , 2 3 + 𝑡 3 4 , 8 7 , 7 6

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