Solved Examples

Example 1: Given three non-collinear points P₁​(x₁​,y₁​,z₁​), P₂​(x₂​,y₂​,z₂​), and P₃​(x₃​,y₃​,z₃​), find the equation of the plane passing through these points.

Solution:

Given three points P₁(1, 2, 3), P₂(4, 5, 6), and P₃(7, 8, 9).

1. Find two vectors on the plane:

[Tex]\vec{v_1}[/Tex] = ❬ 4 – 1, 5 – 2, 6 – 3 ❭ = ❬ 3, 3, 3 }

[Tex]\vec{v_2}[/Tex] = ❬ 7 – 1, 8 – 2, 9 – 3 ❭ = ❬ 6, 6, 6 }

2. Compute the cross product of these vectors to find the normal vector:

[Tex]\vec{n} = \vec{v_1} \times \vec{v_2} [/Tex]

= [Tex]\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 3 & 3 \\ 6 & 6 & 6 \end{vmatrix}[/Tex]

= [Tex]\hat{i}(3*6 – 3*6) – \hat{j}(3*6 – 3*6) + \hat{k}(3*6 – 3*6)[/Tex]

= ❬ 0,0,0 ❭

3. Choose one of the given points, say P₁(1, 2, 3), and use the point-normal form of the equation of the plane:

0(x – 1) + 0(y – 2) + 0(z – 3) = 0

⇒ 0 = 0

The equation simplifies to (0 = 0), which is always true. This means that the points (P₁), (P₂), and (P₃) are collinear, and there’s no unique plane passing through all three points.

Example 2: Given a point P₀​(x₀​,y₀​,z₀​) on the plane and a normal vector [Tex]\vec{n}[/Tex] =⟨a,b,c⟩, find the equation of the plane passing through P₀​ with the given normal vector.

Solution:

Given a point P₀(1, -2, 3) on the plane and a normal vector ( [Tex]\vec{n}[/Tex] = ❬ 2, -1, 4 ❭), let’s find the equation of the plane passing through ( P₀) with the given normal vector.

Using the point-normal form of the equation of a plane:

[Tex]\vec{n} \cdot (\vec{r} – \vec{r_0}) = 0[/Tex]

Substitute the given values:

❬ 2, -1, 4 ❭ · (❬ x, y, z ❭ – ❬ 1, -2, 3 ❭) = 0 \]

⇒ 2(x – 1) – 1(y + 2) + 4(z – 3) = 0

⇒ 2x – 2 – y – 2 + 4z – 12 = 0

⇒ 2x – y + 4z – 16 = 0

Hence, the equation of the plane passing through the point P₀(1, -2, 3) with the normal vector ( [Tex]\vec{n}[/Tex] = ❬ 2, -1, 4 ❭) is:

2x – y + 4z – 16 = 0

Example 3: Given two planes with their equations ax + by + cz + d₁​=0 and ex + fy + gz + d₂​=0, find the equation of the line of intersection between these two planes.

Solution:

Given two planes with equations (2x + y – z + 4 = 0) and (3x – y + 2z – 6 = 0), let’s find the equation of the line of intersection between these two planes.

To find the line of intersection, we will solve the system of equations formed by equating the two planes:

• 2x + y – z + 4 = 0
• 3x – y + 2z – 6 = 0

1. Multiply the first equation by 3 and the second equation by 2 to make the coefficients of (y) equal and opposite:

• 6x + 3y – 3z + 12 = 0
• 6x – 2y + 4z – 12 = 0

2. Subtract the second equation from the first:

(6x + 3y – 3z + 12) – (6x – 2y + 4z – 12) = 0

⇒ 6x + 3y – 3z + 12 – 6x + 2y – 4z + 12 = 0

⇒ 5y – 7z + 24 = 0

3. Rearrange the equation to isolate (y):

5y = 7z – 24

⇒ y = 7/5z – 24/5

4. Express (z) in terms of a parameter (t):

z = t

5. Substitute (z = t) into the equation for (y):

y = 7/5t – 24/5

6. Express (x) in terms of (t) using one of the original equations.

2x + y – z + 4 = 0

⇒ 2x + (7/5t – 24/5) – t + 4 = 0

⇒ 2x + 7/5t – t – 24/5 + 4 = 0

⇒ 2x + 7/5t – 5/5t – 24/5 + 20/5 = 0

⇒ [Tex]2x + \frac{2}{5}t – \frac{24}{5} = 0[/Tex]

⇒ [Tex]2x + \frac{2}{5}t = \frac{24}{5}[/Tex]

⇒ 2x = [Tex]\frac{24}{5} – \frac{2}{5}t[/Tex]

⇒ 2x = [Tex]\frac{24 – 2t}{5}[/Tex]

⇒ x = [Tex]\frac{24 – 2t}{10}[/Tex]

⇒ x = [Tex]\frac{12 – t}{5}[/Tex]

Hence, the parametric equations for the line of intersection between the two planes are:

• x = [Tex]\frac{12 – t}{5}[/Tex]
• y = 7/5t – 24/5
• z = t

Equation of Plane describes its position and orientation in three-dimensional space, typically represented in the form (ax + by + cz + d = 0), where (a), (b), and (c) are coefficients representing the plane’s normal vector, and (d) is the distance from the origin along the normal vector.

In this article, we will learn about the what is the equation of a plane, its definition and general form the equation, the equation of a plane in 3D Space, a Cartesian form of an equation of a plane, the equation of a plane in intercept and parametric form, etc. At the end of this article, you will see some examples of solved problems that will provide a better understanding of the topic.