In this worksheet, we will practice finding the equation of a plane in different forms, such as intercept and parametric forms.

Q1:

Find the equation of the plane whose 𝑥-, 𝑦-, and 𝑧-intercepts are −7, 3, and −4, respectively.

Q2:

Find the general form of the equation of the plane which intersects the coordinate axes at the points (2,0,0), (0,8,0), and (0,0,4).

Q3:

Find, in parametric form, the equation of the plane that passes through the point 𝐴(1,2,1) and contains the two vectors d=⟨1,−1,2⟩ and d=⟨2,−1,1⟩.

Q4:

Which of the following is a parametric form of the equation of the plane that contains the line 𝑥+1−2=𝑦−22=𝑧−54 and the vector d=⟨1,3,1⟩?

Q5:

Which of the following is the parametric form of the equation of the plane that contains the two lines 𝑥−1−2=𝑦+1−1=𝑧−13 and 𝑥−4=𝑦−2−2=𝑧+16?

Q6:

Find the parametric form of the equation of the plane that passes through the points 𝐴(1,5,1), 𝐵(3,4,3), and 𝐶(2,3,4).

Q7:

Write, in intercept form, the equation of the plane 16𝑥+2𝑦+8𝑧−16=0.

Q8:

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

Q9:

What is the length of the segment of the 𝑥-axis cut off by the plane 6𝑥+3𝑦+5𝑧=4?

Q10:

Given that the plane 2𝑥+6𝑦+2𝑧=18 intersects the coordinate axes 𝑥, 𝑦, and 𝑧 at the points 𝐴, 𝐵, and 𝐶, respectively, find the area of △𝐴𝐵𝐶.

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