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).

  • A𝑥+3𝑧+20=0
  • B𝑥+3𝑧20=0
  • C𝑥2𝑦𝑧4=0
  • D𝑥2𝑦𝑧2=0
  • E𝑥2𝑦𝑧+2=0

Q2:

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

Q3:

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

Q4:

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

Q5:

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

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(7,1,13)
  • 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.

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

Q8:

Find the equation of the plane 𝑥𝑦.

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

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

Q10:

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

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.

  • A29𝑥40𝑦+𝑧43=0
  • B4𝑥+3𝑦+4𝑧+146=0
  • C29𝑥+40𝑦𝑧+43=0
  • D7𝑥5𝑦+3𝑧76=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.

  • A𝑥+52=𝑦69=𝑧62
  • B𝑥25=𝑦+96=𝑧+26
  • C𝑥52=𝑦+69=𝑧+62
  • D𝑥+25=𝑦96=𝑧26

Q14:

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

  • A12𝑥9𝑦+18𝑧19=0
  • B4𝑥14𝑦18𝑧+19=0
  • C2𝑥7𝑦9𝑧=0
  • D4𝑥+3𝑦+6𝑧=19

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.

  • A7𝑥7𝑦74𝑧1=0
  • B4𝑥+4𝑦+𝑧+7=0
  • C74𝑥74𝑦7𝑧+1=0
  • D𝑥+𝑦+4𝑧+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.

  • A4𝑥4𝑦+3𝑧10=0
  • B4𝑥4𝑦+3𝑧+10=0
  • C4𝑥4𝑦+3𝑧+4=0
  • D5𝑥+𝑦2𝑧+10=0
  • E5𝑥+𝑦2𝑧10=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).

  • Ar=5,4,1
  • Br=15
  • C(5,4,1)=1,2,2r
  • D5,4,1=15r

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).

  • A4,8,3=3r
  • B4,4,3=1r
  • C4,8,3=3r
  • D20,16,9=23r

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)?

  • A𝑥+𝑦𝑧+5=0
  • B𝑥𝑦𝑧5=0
  • C𝑥𝑦+𝑧+5=0
  • D𝑥+𝑦+𝑧5=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.

  • A25𝑥11𝑦13𝑧+81=0
  • B3𝑥8𝑦+𝑧58=0
  • C7𝑥5𝑦3𝑧+11=0
  • D2𝑥+𝑦+3𝑧+28=0
  • E𝑥7𝑦4𝑧+30=0

Q22:

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

  • A14𝑥10𝑦+8𝑧+56=0
  • B14𝑥10𝑦+8𝑧+8=0
  • C3𝑥+𝑦3𝑧8=0
  • D14𝑥10𝑦+8𝑧8=0
  • E3𝑥+𝑦3𝑧+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).

  • A7𝑥+4𝑦+4𝑧52=0
  • B4𝑥3𝑦+4𝑧16=0
  • C𝑥8𝑦3𝑧15=0
  • D7𝑥+4𝑦+4𝑧+51=0
  • E4𝑥+3𝑦+4𝑧+32=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.

  • A6𝑥+9𝑦+8𝑧77=0
  • B2𝑥+𝑦+3𝑧21=0
  • C𝑥7𝑦+3𝑧+33=0
  • D𝑥+7𝑦+3𝑧93=0
  • E2𝑥+𝑦+3𝑧+17=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).

  • A6𝑥6𝑦𝑧75=0
  • B6𝑥+6𝑦𝑧+75=0
  • C7𝑥𝑦+4𝑧32=0
  • D7𝑥+𝑦4𝑧+32=0
  • E𝑥+7𝑦+5𝑧+43=0

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