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

In this lesson, we will learn how to find the equation of a plane in different forms like the general, vector, and parametric forms.

Worksheet: The Equation of a Plane in 3D in Different Forms • 25 Questions

Q1:

Find the general equation of the plane 𝑥 = 4 + 7 𝑡 + 4 𝑡 1 2 , 𝑦 = 3 4 𝑡 2 , 𝑧 = 1 + 3 𝑡 1 .

Q2:

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

Q3:

Which of the following points lies in the plane 3 ( 𝑥 + 4 ) 2 ( 𝑦 + 1 ) 7 ( 𝑧 6 ) = 0 ?

Q4:

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

Q5:

Write, in intercept form, the equation of the plane 1 6 𝑥 + 2 𝑦 + 8 𝑧 1 6 = 0 .

Q6:

Find the equation of the plane 𝑥 𝑦 .

Q7:

Find the equation of the plane which is perpendicular to the vector 𝐴 = 5 𝑖 7 𝑗 3 𝑘 and passes through the point 𝐵 ( 5 , 5 , 9 ) .

Q8:

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

Q9:

Which of the following does the equation 7 𝑥 2 𝑧 = 0 represent in three-dimensional space?

Q10:

Determine the general form of the equation for a plane in which the two straight lines 𝐿 𝑥 + 8 7 = 𝑦 + 7 5 = 𝑧 + 5 3 1 and 𝐿 𝑥 + 8 4 = 𝑦 + 7 3 = 𝑧 + 5 4 2 lie.

Q11:

Determine the Cartesian equation of the straight line passing through the point ( 2 , 9 , 2 ) that is perpendicular to the plane 5 𝑥 6 𝑦 6 𝑧 1 1 = 0 .

Q12:

To which of the following planes is the straight line 𝑥 2 4 = 𝑦 + 7 3 = 𝑧 + 9 6 perpendicular?

Q13:

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.

Q14:

Determine the general equation of the plane that contains the straight line 𝑥 + 2 7 = 𝑦 6 5 = 𝑧 + 9 5 and that is perpendicular to the plane 𝑥 + 𝑦 2 𝑧 = 2 .

Q15:

Find the general equation of a plane that is parallel to the 𝑥 -axis.

Q16:

Find the general equation of a plane that is parallel to the 𝑧 -axis.

Q17:

Find the general equation of a plane that is parallel to the 𝑦 -axis.

Q18:

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

Q19:

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

Q20:

Find the vector form of the equation of the plane containing the two straight lines 𝑟 = ( 𝑖 𝑗 3 𝑘 ) + 𝑡 ( 3 𝑖 + 3 𝑗 + 4 𝑘 ) 1 1 and 𝑟 = ( 𝑖 2 𝑗 3 𝑘 ) + 𝑡 ( 𝑖 2 𝑗 4 𝑘 ) 2 2 .

Q21:

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 ) ?

Q22:

Given that 𝐴 𝐵 is parallel to the plane 8 𝑥 5 𝑦 2 𝑧 5 = 0 , where the coordinates of 𝐴 and 𝐵 are ( 4 , 3 , 𝑚 ) and ( 3 , 3 , 𝑛 ) , respectively, find the value of ( 𝑛 𝑚 ) .

Q23:

Find the general equation of the plane which passes through the point ( 2 , 8 , 1 ) and is perpendicular to the two planes 6 𝑥 4 𝑦 + 6 𝑧 = 5 and 5 𝑥 + 3 𝑦 6 𝑧 = 3 .

Q24:

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

Q25:

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

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