Worksheet: The Equation of a Plane in 3D in Different Forms

In this worksheet, we will practice finding the equation of a plane in different forms like the general, vector, and parametric forms.

Q1:

Find the general equation of the plane ๐‘ฅ = 4 + 7 ๐‘ก + 4 ๐‘ก ๏Šง ๏Šจ , ๐‘ฆ = โˆ’ 3 โˆ’ 4 ๐‘ก ๏Šจ , ๐‘ง = 1 + 3 ๐‘ก ๏Šง .

  • A ๐‘ฅ + 1 2 ๐‘ฆ โˆ’ 2 8 ๐‘ง = 0
  • B ๐‘ฅ โˆ’ 1 2 ๐‘ฆ + 2 8 ๐‘ง โˆ’ 1 6 = 0
  • C ๐‘ฅ + 4 ๐‘ฆ + 7 ๐‘ง + 1 6 = 0
  • D 3 ๐‘ฅ + 3 ๐‘ฆ โˆ’ 7 ๐‘ง + 4 = 0
  • E 1 2 ๐‘ฅ + 4 ๐‘ฆ + 7 ๐‘ง โˆ’ 4 3 = 0

Q2:

In which of the following planes does the point ( 3 , โˆ’ 1 , 5 ) lie?

  • A 2 ๐‘ฅ + ๐‘ฆ โˆ’ 2 ๐‘ง + 2 3 = 0
  • B โˆ’ 4 ๐‘ฅ โˆ’ 4 ๐‘ฆ + 2 ๐‘ง + 7 = 0
  • C 2 ๐‘ฅ โˆ’ 4 ๐‘ฆ + ๐‘ง + 5 = 0
  • D ๐‘ฅ โˆ’ 2 ๐‘ฆ + 2 ๐‘ง โˆ’ 1 5 = 0
  • E 3 ๐‘ฅ โˆ’ ๐‘ฆ + 5 ๐‘ง = 0

Q3:

Which of the following points lies in the plane 3 ( ๐‘ฅ + 4 ) โˆ’ 2 ( ๐‘ฆ + 1 ) โˆ’ 7 ( ๐‘ง โˆ’ 6 ) = 0 ?

  • A ( 7 , โˆ’ 1 , โˆ’ 1 3 )
  • B ( 3 , โˆ’ 2 , โˆ’ 7 )
  • C ( 4 , 1 , โˆ’ 6 )
  • D ( โˆ’ 4 , โˆ’ 1 , 6 )

Q4:

Find the general equation of the plane which passes through the point ( 3 , โˆ’ 8 , โˆ’ 7 ) and contains the ๐‘ฅ -axis.

  • A 8 ๐‘ฅ โˆ’ 7 ๐‘ฆ = 0
  • B โˆ’ 7 ๐‘ฅ + 8 ๐‘ง = 0
  • C 3 ๐‘ฅ โˆ’ 7 ๐‘ฆ + 8 ๐‘ง = 0
  • D โˆ’ 7 ๐‘ฆ + 8 ๐‘ง = 0
  • E 3 ๐‘ฅ โˆ’ 8 ๐‘ฆ โˆ’ 7 ๐‘ง = 0

Q5:

Write, in intercept form, the equation of the plane 1 6 ๐‘ฅ + 2 ๐‘ฆ + 8 ๐‘ง โˆ’ 1 6 = 0 .

  • A ๐‘ฅ 1 6 + ๐‘ฆ 2 + ๐‘ง 8 = 1
  • B ๐‘ฅ 1 + ๐‘ฆ 8 + ๐‘ง 2 = 1 6
  • C ๐‘ฅ 1 6 + ๐‘ฆ 2 + ๐‘ง 8 = 1 6
  • D ๐‘ฅ 1 + ๐‘ฆ 8 + ๐‘ง 2 = 1
  • E ๐‘ฅ 1 6 + ๐‘ฆ 1 6 + ๐‘ง 1 6 = 1

Q6:

Find the equation of the plane ๐‘ฅ ๐‘ฆ .

  • A ๐‘ฅ + ๐‘ฆ = ๐‘ง
  • B ๐‘ฅ + ๐‘ฆ = 0
  • C ๐‘ฅ = ๐‘ฆ
  • D ๐‘ง = 0
  • E ๐‘ง โˆ’ ๐‘ฅ ๐‘ฆ = 0

Q7:

Find the equation of the plane which is perpendicular to the vector A i j k = 5 โˆ’ 7 โˆ’ 3 and passes through the point ๐ต ( โˆ’ 5 , 5 , 9 ) .

  • A โˆ’ 5 ๐‘ฅ + 5 ๐‘ฆ + 9 ๐‘ง โˆ’ 5 = 0
  • B 5 ๐‘ฅ โˆ’ 7 ๐‘ฆ โˆ’ 3 ๐‘ง โˆ’ 8 7 = 0
  • C 5 ๐‘ฅ โˆ’ 7 ๐‘ฆ โˆ’ 3 ๐‘ง โˆ’ 5 = 0
  • D 5 ๐‘ฅ โˆ’ 7 ๐‘ฆ โˆ’ 3 ๐‘ง + 8 7 = 0
  • E โˆ’ 5 ๐‘ฅ + 5 ๐‘ฆ + 9 ๐‘ง + 8 7 = 0

Q8:

A plane passes through ( โˆ’ 2 , โˆ’ 2 , 3 ) and has normal โŸจ โˆ’ 4 , 1 , โˆ’ 4 โŸฉ . Give its equation in vector form.

  • A r = โˆ’ 6
  • B โŸจ โˆ’ 4 , 1 , โˆ’ 4 โŸฉ โ‹… = โŸจ โˆ’ 2 , โˆ’ 2 , 3 โŸฉ r
  • C r = โŸจ โˆ’ 4 , 1 , โˆ’ 4 โŸฉ
  • D โŸจ โˆ’ 4 , 1 , โˆ’ 4 โŸฉ โ‹… = โˆ’ 6 r

Q9:

Which of the following does the equation โˆ’ 7 ๐‘ฅ โˆ’ 2 ๐‘ง = 0 represent in three-dimensional space?

  • Aa plane containing the ๐‘ง -axis
  • Ba plane containing the ๐‘ฅ -axis
  • Ca straight line whose direction ratios are ( โˆ’ 7 , 0 , โˆ’ 2 )
  • Da plane containing the ๐‘ฆ -axis

Q10:

Determine the general form of the equation for a plane in which the two straight lines ๐ฟ โˆถ ๐‘ฅ + 8 โˆ’ 7 = ๐‘ฆ + 7 โˆ’ 5 = ๐‘ง + 5 3 ๏Šง and ๐ฟ โˆถ ๐‘ฅ + 8 4 = ๐‘ฆ + 7 3 = ๐‘ง + 5 4 ๏Šจ lie.

  • A โˆ’ 7 ๐‘ฅ โˆ’ 5 ๐‘ฆ + 3 ๐‘ง โˆ’ 7 6 = 0
  • B โˆ’ 2 9 ๐‘ฅ โˆ’ 4 0 ๐‘ฆ + ๐‘ง โˆ’ 4 3 = 0
  • C 4 ๐‘ฅ + 3 ๐‘ฆ + 4 ๐‘ง + 1 4 6 = 0
  • D โˆ’ 2 9 ๐‘ฅ + 4 0 ๐‘ฆ โˆ’ ๐‘ง + 4 3 = 0

Q11:

Determine the Cartesian equation of the straight line passing through the point ( โˆ’ 2 , 9 , 2 ) that is perpendicular to the plane 5 ๐‘ฅ โˆ’ 6 ๐‘ฆ โˆ’ 6 ๐‘ง โˆ’ 1 1 = 0 .

  • A ๐‘ฅ โˆ’ 5 โˆ’ 2 = ๐‘ฆ + 6 9 = ๐‘ง + 6 2
  • B ๐‘ฅ โˆ’ 2 5 = ๐‘ฆ + 9 โˆ’ 6 = ๐‘ง + 2 โˆ’ 6
  • C ๐‘ฅ + 5 โˆ’ 2 = ๐‘ฆ โˆ’ 6 9 = ๐‘ง โˆ’ 6 2
  • D ๐‘ฅ + 2 5 = ๐‘ฆ โˆ’ 9 โˆ’ 6 = ๐‘ง โˆ’ 2 โˆ’ 6

Q12:

To which of the following planes is the straight line ๐‘ฅ โˆ’ 2 4 = ๐‘ฆ + 7 โˆ’ 3 = ๐‘ง + 9 6 perpendicular?

  • A 2 ๐‘ฅ โˆ’ 7 ๐‘ฆ โˆ’ 9 ๐‘ง = 0
  • B 4 ๐‘ฅ โˆ’ 1 4 ๐‘ฆ โˆ’ 1 8 ๐‘ง + 1 9 = 0
  • C 4 ๐‘ฅ + 3 ๐‘ฆ + 6 ๐‘ง = โˆ’ 1 9
  • D 1 2 ๐‘ฅ โˆ’ 9 ๐‘ฆ + 1 8 ๐‘ง โˆ’ 1 9 = 0

Q13:

Find the general equation of the plane which passes through the two points ๐ด ( 8 , โˆ’ 7 , โˆ’ 2 ) and ๐ต ( 1 , โˆ’ 4 , โˆ’ 1 ) , given that the distance from the ๐‘ฅ -intercept to the origin is equal to the distance from the ๐‘ฆ -intercept to the origin.

  • A 4 ๐‘ฅ + 4 ๐‘ฆ + ๐‘ง + 7 = 0
  • B โˆ’ 7 ๐‘ฅ โˆ’ 7 ๐‘ฆ โˆ’ 7 4 ๐‘ง โˆ’ 1 = 0
  • C โˆ’ 7 4 ๐‘ฅ โˆ’ 7 4 ๐‘ฆ โˆ’ 7 ๐‘ง + 1 = 0
  • D ๐‘ฅ + ๐‘ฆ + 4 ๐‘ง + 7 = 0

Q14:

Determine the general equation of the plane that contains the straight line ๐‘ฅ + 2 7 = ๐‘ฆ โˆ’ 6 5 = ๐‘ง + 9 5 and that is perpendicular to the plane โˆ’ ๐‘ฅ + ๐‘ฆ โˆ’ 2 ๐‘ง = 2 .

  • A 7 ๐‘ฅ + 5 ๐‘ฆ + 5 ๐‘ง + 2 9 = 0
  • B 5 ๐‘ฅ + 3 ๐‘ฆ โˆ’ 4 ๐‘ง โˆ’ 4 4 = 0
  • C 7 ๐‘ฅ + 5 ๐‘ฆ + 5 ๐‘ง โˆ’ 2 6 = 0
  • D 5 ๐‘ฅ โˆ’ 3 ๐‘ฆ โˆ’ 4 ๐‘ง โˆ’ 8 = 0
  • E โˆ’ 2 ๐‘ฅ + 6 ๐‘ฆ โˆ’ 9 ๐‘ง + 1 2 = 0

Q15:

Find the general equation of a plane that is parallel to the ๐‘ฅ -axis.

  • A ๐‘Ž ๐‘ฅ + ๐‘ ๐‘ฆ + ๐‘‘ = 0
  • B ๐‘Ž ๐‘ฆ + ๐‘ ๐‘ง + ๐‘‘ = ๐‘ฅ
  • C ๐‘Ž ๐‘ฅ + ๐‘ ๐‘ฆ + ๐‘ ๐‘ง + ๐‘‘ = 0
  • D ๐‘ ๐‘ฆ + ๐‘ ๐‘ง + ๐‘‘ = 0
  • E ๐‘ ๐‘ฆ + ๐‘ ๐‘ง + ๐‘‘ = ๐‘ฅ

Q16:

Find the general equation of a plane that is parallel to the ๐‘ง -axis.

  • A ๐‘ ๐‘ฆ + ๐‘ ๐‘ง + ๐‘‘ = ๐‘ฅ
  • B ๐‘Ž ๐‘ฅ + ๐‘ ๐‘ง + ๐‘‘ = 0
  • C ๐‘Ž ๐‘ฅ + ๐‘ ๐‘ฆ + ๐‘ ๐‘ง + ๐‘‘ = 0
  • D ๐‘Ž ๐‘ฅ + ๐‘ ๐‘ฆ + ๐‘‘ = 0
  • E ๐‘Ž ๐‘ฅ + ๐‘ ๐‘ฆ + ๐‘‘ = ๐‘ง

Q17:

Find the general equation of a plane that is parallel to the ๐‘ฆ -axis.

  • A ๐‘ ๐‘ฆ + ๐‘ ๐‘ง + ๐‘‘ = ๐‘ฅ
  • B ๐‘Ž ๐‘ฅ + ๐‘ ๐‘ฆ + ๐‘‘ = 0
  • C ๐‘Ž ๐‘ฅ + ๐‘ ๐‘ฆ + ๐‘ ๐‘ง + ๐‘‘ = 0
  • D ๐‘Ž ๐‘ฅ + ๐‘ ๐‘ง + ๐‘‘ = 0
  • E ๐‘Ž ๐‘ฅ + ๐‘ ๐‘ง + ๐‘‘ = ๐‘ฆ

Q18:

Write, in normal form, the equation of the plane containing the point and perpendicular to the vector .

  • A
  • B
  • C
  • D
  • E

Q19:

Find the equation, in vector form, of the plane passing through the points ( 1 , 2 , 2 ) , ( 3 , 1 , โˆ’ 4 ) , and ( 0 , 3 , 3 ) .

  • A ( 5 , 4 , 1 ) โ‹… = โŸจ 1 , 2 , 2 โŸฉ r
  • B r = 1 5
  • C r = โŸจ 5 , 4 , 1 โŸฉ
  • D โŸจ 5 , 4 , 1 โŸฉ โ‹… = 1 5 r

Q20:

Find the vector form of the equation of the plane containing the two straight lines r i j k i j k ๏Šง ๏Šง = ( โˆ’ โˆ’ 3 ) + ๐‘ก ( 3 + 3 + 4 ) and r i j k i j k ๏Šจ ๏Šจ = ( โˆ’ โˆ’ 2 โˆ’ 3 ) + ๐‘ก ( โˆ’ โˆ’ 2 โˆ’ 4 ) .

  • A โŸจ 2 0 , 1 6 , 9 โŸฉ โ‹… = โˆ’ 2 3 r
  • B โŸจ 4 , โˆ’ 4 , 3 โŸฉ โ‹… = โˆ’ 1 r
  • C โŸจ 4 , โˆ’ 8 , 3 โŸฉ โ‹… = โˆ’ 3 r
  • D โŸจ 4 , โˆ’ 8 , 3 โŸฉ โ‹… = 3 r

Q21:

Which of the following is the equation of a plane that bisects the line segment between the two points ( 4 , โˆ’ 2 , โˆ’ 6 ) and ( 8 , 4 , 2 ) ?

  • A ๐‘ฅ โˆ’ ๐‘ฆ โˆ’ ๐‘ง โˆ’ 5 = 0
  • B ๐‘ฅ โˆ’ ๐‘ฆ + ๐‘ง + 5 = 0
  • C ๐‘ฅ + ๐‘ฆ โˆ’ ๐‘ง + 5 = 0
  • D ๐‘ฅ + ๐‘ฆ + ๐‘ง โˆ’ 5 = 0

Q22:

Given that โƒ– ๏ƒฉ ๏ƒฉ ๏ƒฉ ๏ƒฉ โƒ— ๐ด ๐ต is parallel to the plane 8 ๐‘ฅ โˆ’ 5 ๐‘ฆ โˆ’ 2 ๐‘ง โˆ’ 5 = 0 , where the coordinates of ๐ด and ๐ต are ( โˆ’ 4 , 3 , ๐‘š ) and ( โˆ’ 3 , โˆ’ 3 , ๐‘› ) , respectively, find the value of ( ๐‘› โˆ’ ๐‘š ) .

Q23:

Find the general equation of the plane which passes through the point ( 2 , 8 , 1 ) and is perpendicular to the two planes โˆ’ 6 ๐‘ฅ โˆ’ 4 ๐‘ฆ + 6 ๐‘ง = โˆ’ 5 and 5 ๐‘ฅ + 3 ๐‘ฆ โˆ’ 6 ๐‘ง = 3 .

  • A โˆ’ 3 ๐‘ฅ โˆ’ 2 ๐‘ฆ + 3 ๐‘ง + 1 9 = 0
  • B 3 ๐‘ฅ + 3 ๐‘ฆ + ๐‘ง โˆ’ 3 1 = 0
  • C 5 ๐‘ฅ + 3 ๐‘ฆ โˆ’ 6 ๐‘ง โˆ’ 2 8 = 0
  • D 3 ๐‘ฅ โˆ’ 3 ๐‘ฆ + ๐‘ง + 1 7 = 0
  • E 2 ๐‘ฅ + 8 ๐‘ฆ + ๐‘ง + 7 8 = 0

Q24:

Find the Cartesian equation of the plane ( ๐‘ฅ , ๐‘ฆ , ๐‘ง ) = ( โˆ’ 7 , โˆ’ 5 , โˆ’ 3 ) + ๐‘ก ( โˆ’ 3 , โˆ’ 8 , 1 ) + ๐‘ก ( 2 , 1 , 3 ) ๏Šง ๏Šจ , where ๐‘ก ๏Šง and ๐‘ก ๏Šจ are parameters.

  • A โˆ’ 3 ๐‘ฅ โˆ’ 8 ๐‘ฆ + ๐‘ง โˆ’ 5 8 = 0
  • B ๐‘ฅ โˆ’ 7 ๐‘ฆ โˆ’ 4 ๐‘ง + 3 0 = 0
  • C 2 ๐‘ฅ + ๐‘ฆ + 3 ๐‘ง + 2 8 = 0
  • D 2 5 ๐‘ฅ โˆ’ 1 1 ๐‘ฆ โˆ’ 1 3 ๐‘ง + 8 1 = 0
  • E โˆ’ 7 ๐‘ฅ โˆ’ 5 ๐‘ฆ โˆ’ 3 ๐‘ง + 1 1 = 0

Q25:

Write, in normal form, the equation of the plane containing ( โˆ’ 3 , 1 , โˆ’ 3 ) , ( 4 , โˆ’ 4 , 3 ) , and ( 0 , 0 , 1 ) .

  • A โˆ’ 1 4 ๐‘ฅ โˆ’ 1 0 ๐‘ฆ + 8 ๐‘ง + 8 = 0
  • B โˆ’ 3 ๐‘ฅ + ๐‘ฆ โˆ’ 3 ๐‘ง โˆ’ 8 = 0
  • C โˆ’ 3 ๐‘ฅ + ๐‘ฆ โˆ’ 3 ๐‘ง + 8 = 0
  • D โˆ’ 1 4 ๐‘ฅ โˆ’ 1 0 ๐‘ฆ + 8 ๐‘ง โˆ’ 8 = 0
  • E โˆ’ 1 4 ๐‘ฅ โˆ’ 1 0 ๐‘ฆ + 8 ๐‘ง + 5 6 = 0

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