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

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.


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

  • Arijk(26+15)=31
  • Brijk(26+15)=0
  • Crijk(23+5)=29
  • Drijk(23+5)=58
  • Erijk(4+)=0


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

  • A1,1,1=14r
  • Br=14
  • C1,1,1=2,6,6r
  • Dr=2,6,6


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

  • A48989,88989,38989
  • B489,889,389
  • C41515,81515,155
  • D14,12,316


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

  • A2𝑥+𝑦2𝑧+23=0
  • B𝑥2𝑦+2𝑧15=0
  • C2𝑥4𝑦+𝑧+5=0
  • D4𝑥4𝑦+2𝑧+7=0
  • E3𝑥𝑦+5𝑧=0


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(7,1,13)
  • D(4,1,6)


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

  • A3𝑥7𝑦+8𝑧=0
  • B3𝑥8𝑦7𝑧=0
  • C7𝑥+8𝑧=0
  • D8𝑥7𝑦=0
  • E7𝑦+8𝑧=0


Find the equation of the plane 𝑥𝑦.

  • A𝑥+𝑦=0
  • B𝑥+𝑦=𝑧
  • C𝑧=0
  • D𝑧𝑥𝑦=0
  • E𝑥=𝑦


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

  • A5𝑥+5𝑦+9𝑧5=0
  • B5𝑥7𝑦3𝑧+87=0
  • C5𝑥+5𝑦+9𝑧+87=0
  • D5𝑥7𝑦3𝑧5=0
  • E5𝑥7𝑦3𝑧87=0


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

  • Ar=6
  • Br=4,1,4
  • C4,1,4=6r
  • D4,1,4=2,2,3r


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

This lesson includes 43 additional questions and 382 additional question variations for subscribers.

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