Worksheet: Equation of a Plane: Vector, Scalar, and General Forms

In this worksheet, we will practice finding the vector, scalar (standard or component), and general (Cartesian or normal) forms of the equation of a plane given the normal vector and a point on it.

Q1:

Write, in normal form, the equation of the plane (1,0,3), (1,2,1) and (6,1,6).

  • A 𝑥 + 3 𝑧 + 2 0 = 0
  • B 𝑥 + 3 𝑧 2 0 = 0
  • C 𝑥 2 𝑦 𝑧 4 = 0
  • D 𝑥 2 𝑦 𝑧 2 = 0
  • E 𝑥 2 𝑦 𝑧 + 2 = 0

Q2:

Which of the following planes contains the straight line rijkijk=64+𝑡(4+)?

  • A r i j k ( 2 6 + 1 5 ) = 3 1
  • B r i j k ( 2 6 + 1 5 ) = 0
  • C r i j k ( 2 3 + 5 ) = 2 9
  • D r i j k ( 2 3 + 5 ) = 5 8
  • E r i j k ( 4 + ) = 0

Q3:

Find the vector form of the equation of the plane that has normal vector nijk=++ and contains the point (2,6,6).

  • A 1 , 1 , 1 = 1 4 r
  • B r = 1 4
  • C 1 , 1 , 1 = 2 , 6 , 6 r
  • D r = 2 , 6 , 6

Q4:

Find the direction cosines of the normal to the plane 4𝑥+8𝑦3𝑧=28.

  • A 4 8 9 8 9 , 8 8 9 8 9 , 3 8 9 8 9
  • B 4 8 9 , 8 8 9 , 3 8 9
  • C 4 1 5 1 5 , 8 1 5 1 5 , 1 5 5
  • D 1 4 , 1 2 , 3 1 6

Q5:

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

  • A 2 𝑥 + 𝑦 2 𝑧 + 2 3 = 0
  • B 𝑥 2 𝑦 + 2 𝑧 1 5 = 0
  • C 2 𝑥 4 𝑦 + 𝑧 + 5 = 0
  • D 4 𝑥 4 𝑦 + 2 𝑧 + 7 = 0
  • E 3 𝑥 𝑦 + 5 𝑧 = 0

Q6:

Which of the following points lies in the plane 3(𝑥+4)2(𝑦+1)7(𝑧6)=0?

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

Q7:

Find the general equation of the plane which passes through the point (3,8,7) and contains the 𝑥-axis.

  • A 3 𝑥 7 𝑦 + 8 𝑧 = 0
  • B 3 𝑥 8 𝑦 7 𝑧 = 0
  • C 7 𝑥 + 8 𝑧 = 0
  • D 8 𝑥 7 𝑦 = 0
  • E 7 𝑦 + 8 𝑧 = 0

Q8:

Find the equation of the plane 𝑥𝑦.

  • A 𝑥 + 𝑦 = 0
  • B 𝑥 + 𝑦 = 𝑧
  • C 𝑧 = 0
  • D 𝑧 𝑥 𝑦 = 0
  • E 𝑥 = 𝑦

Q9:

Find the equation of the plane which is perpendicular to the vector Aijk=573 and passes through the point 𝐵(5,5,9).

  • A 5 𝑥 + 5 𝑦 + 9 𝑧 5 = 0
  • B 5 𝑥 7 𝑦 3 𝑧 + 8 7 = 0
  • C 5 𝑥 + 5 𝑦 + 9 𝑧 + 8 7 = 0
  • D 5 𝑥 7 𝑦 3 𝑧 5 = 0
  • E 5 𝑥 7 𝑦 3 𝑧 8 7 = 0

Q10:

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

  • A r = 6
  • B 4 , 1 , 4 = 2 , 2 , 3 r
  • C 4 , 1 , 4 = 6 r
  • D r = 4 , 1 , 4

Q11:

Which of the following does the equation 7𝑥2𝑧=0 represent in three-dimensional space?

  • Aa plane containing the 𝑦-axis
  • Ba plane containing the 𝑧-axis
  • Ca straight line whose direction ratios are (7,0,2)
  • Da plane containing the 𝑥-axis

Q12:

Determine the general form of the equation for a plane in which the two straight lines 𝐿𝑥+87=𝑦+75=𝑧+53 and 𝐿𝑥+84=𝑦+73=𝑧+54 lie.

  • A 2 9 𝑥 4 0 𝑦 + 𝑧 4 3 = 0
  • B 4 𝑥 + 3 𝑦 + 4 𝑧 + 1 4 6 = 0
  • C 2 9 𝑥 + 4 0 𝑦 𝑧 + 4 3 = 0
  • D 7 𝑥 5 𝑦 + 3 𝑧 7 6 = 0

Q13:

Determine the Cartesian equation of the straight line passing through the point (2,9,2) that is perpendicular to the plane 5𝑥6𝑦6𝑧11=0.

  • A 𝑥 + 5 2 = 𝑦 6 9 = 𝑧 6 2
  • B 𝑥 2 5 = 𝑦 + 9 6 = 𝑧 + 2 6
  • C 𝑥 5 2 = 𝑦 + 6 9 = 𝑧 + 6 2
  • D 𝑥 + 2 5 = 𝑦 9 6 = 𝑧 2 6

Q14:

To which of the following planes is the straight line 𝑥24=𝑦+73=𝑧+96 perpendicular?

  • A 1 2 𝑥 9 𝑦 + 1 8 𝑧 1 9 = 0
  • B 4 𝑥 1 4 𝑦 1 8 𝑧 + 1 9 = 0
  • C 2 𝑥 7 𝑦 9 𝑧 = 0
  • D 4 𝑥 + 3 𝑦 + 6 𝑧 = 1 9

Q15:

Find the general equation of the plane which passes through the two points 𝐴(8,7,2) and 𝐵(1,4,1), given that the distance from the 𝑥-intercept to the origin is equal to the distance from the 𝑦-intercept to the origin.

  • A 7 𝑥 7 𝑦 7 4 𝑧 1 = 0
  • B 4 𝑥 + 4 𝑦 + 𝑧 + 7 = 0
  • C 7 4 𝑥 7 4 𝑦 7 𝑧 + 1 = 0
  • D 𝑥 + 𝑦 + 4 𝑧 + 7 = 0

Q16:

Write, in normal form, the equation of the plane P containing the point Q=5,1,2 and perpendicular to the vector n=4,4,3.

  • A 4 𝑥 4 𝑦 + 3 𝑧 1 0 = 0
  • B 4 𝑥 4 𝑦 + 3 𝑧 + 1 0 = 0
  • C 4 𝑥 4 𝑦 + 3 𝑧 + 4 = 0
  • D 5 𝑥 + 𝑦 2 𝑧 + 1 0 = 0
  • E 5 𝑥 + 𝑦 2 𝑧 1 0 = 0

Q17:

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

  • A r = 5 , 4 , 1
  • B r = 1 5
  • C ( 5 , 4 , 1 ) = 1 , 2 , 2 r
  • D 5 , 4 , 1 = 1 5 r

Q18:

Find the vector form of the equation of the plane containing the two straight lines rijkijk=(3)+𝑡(3+3+4) and rijkijk=(23)+𝑡(24).

  • A 4 , 8 , 3 = 3 r
  • B 4 , 4 , 3 = 1 r
  • C 4 , 8 , 3 = 3 r
  • D 2 0 , 1 6 , 9 = 2 3 r

Q19:

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

  • A 𝑥 + 𝑦 𝑧 + 5 = 0
  • B 𝑥 𝑦 𝑧 5 = 0
  • C 𝑥 𝑦 + 𝑧 + 5 = 0
  • D 𝑥 + 𝑦 + 𝑧 5 = 0

Q20:

Given that 𝐴𝐵 is parallel to the plane 8𝑥5𝑦2𝑧5=0, where the coordinates of 𝐴 and 𝐵 are (4,3,𝑚) and (3,3,𝑛), respectively, find the value of (𝑛𝑚).

Q21:

Find the Cartesian equation of the plane (𝑥,𝑦,𝑧)=(7,5,3)+𝑡(3,8,1)+𝑡(2,1,3), where 𝑡 and 𝑡 are parameters.

  • A 2 5 𝑥 1 1 𝑦 1 3 𝑧 + 8 1 = 0
  • B 3 𝑥 8 𝑦 + 𝑧 5 8 = 0
  • C 7 𝑥 5 𝑦 3 𝑧 + 1 1 = 0
  • D 2 𝑥 + 𝑦 + 3 𝑧 + 2 8 = 0
  • E 𝑥 7 𝑦 4 𝑧 + 3 0 = 0

Q22:

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

  • A 1 4 𝑥 1 0 𝑦 + 8 𝑧 + 5 6 = 0
  • B 1 4 𝑥 1 0 𝑦 + 8 𝑧 + 8 = 0
  • C 3 𝑥 + 𝑦 3 𝑧 8 = 0
  • D 1 4 𝑥 1 0 𝑦 + 8 𝑧 8 = 0
  • E 3 𝑥 + 𝑦 3 𝑧 + 8 = 0

Q23:

Find the general equation of the plane which contains the straight line 𝑥17=𝑦+84=𝑧+34 and the point 𝐴(8,4,3).

  • A 7 𝑥 + 4 𝑦 + 4 𝑧 5 2 = 0
  • B 4 𝑥 3 𝑦 + 4 𝑧 1 6 = 0
  • C 𝑥 8 𝑦 3 𝑧 1 5 = 0
  • D 7 𝑥 + 4 𝑦 + 4 𝑧 + 5 1 = 0
  • E 4 𝑥 + 3 𝑦 + 4 𝑧 + 3 2 = 0

Q24:

Find the general equation of the plane which passes through the two points (6,9,8) and (8,5,2) and is parallel to the vector A=2,1,3.

  • A 6 𝑥 + 9 𝑦 + 8 𝑧 7 7 = 0
  • B 2 𝑥 + 𝑦 + 3 𝑧 2 1 = 0
  • C 𝑥 7 𝑦 + 3 𝑧 + 3 3 = 0
  • D 𝑥 + 7 𝑦 + 3 𝑧 9 3 = 0
  • E 2 𝑥 + 𝑦 + 3 𝑧 + 1 7 = 0

Q25:

Find the general equation of the plane passing through the point 𝐴(7,5,3) and perpendicular to the straight line passing through the two points 𝐵(5,7,1) and 𝐶(9,9,7).

  • A 6 𝑥 6 𝑦 𝑧 7 5 = 0
  • B 6 𝑥 + 6 𝑦 𝑧 + 7 5 = 0
  • C 7 𝑥 𝑦 + 4 𝑧 3 2 = 0
  • D 7 𝑥 + 𝑦 4 𝑧 + 3 2 = 0
  • E 𝑥 + 7 𝑦 + 5 𝑧 + 4 3 = 0

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