Q 3. - Calculus

Question

Let $ax+by+cz = d$ be the equation of a plane with $a, b, c$ and d all nonzero. What are the coordinates of the intersection of the plane and the $x-, y-$ and $z-$axes ? Explain how to use these points to sketch the plane.

Step-by-Step Solution

Answer

Coordinates of the intersection of the plane and the $x-$axes is $\left(\frac{d}{a},0,0\right)$

Coordinates of the intersection of the plane and the $y-$axes is $\left(0,\frac{d}{b},0\right)$

Coordinates of intersection of the plane and the $z-$axes is $\left(0,0,\frac{d}{c}\right)$

Step 1: Given information

The equation of the plane to be $a x+b y+c z=d$ with $a, b, c$ and $d$ to be non-zero.

Step 2: Calculation

The goal is to determine the coordinates of the plane's intersection with the $x,y$ and $z$ axes, as well as how to sketch the plane using the points.

When the $y$ and $z$ coordinates are zero, a plane and the $x-$axes intersect.

The coordinates of intersection of the plane and the $x-$axes is obtained by substituting $y=0$ and $z=0$

The substitution of $y=0$ and $z=0$ in $a x+b y+c z=d$ gives:

$$\begin{gathered} ax+b\left(0\right)+c\left(0\right)=d \\ ax=d \\ x=\frac{d}{a} \end{gathered}$$

Therefore, coordinates of intersection of the plane and the $x-$axes is $\left(\frac{d}{a},0,0\right)$

Step 3: Calculation

Substituting $x=0$ and $z=0$ yields the intersection coordinates of the plane and the $y-$axes.

The substitution of $x=0$ and $z=0$ in $a x+b y+c z=d$ gives:

$$\begin{gathered} a\left(0\right)+by+c\left(0\right)=d \\ by=d \\ y=\frac{d}{b} \end{gathered}$$

Therefore, coordinates of intersection of the plane and the $y-$axes is $\left(0,\frac{d}{b},0\right)$

Step 4: Calculation

Substituting $x=0$ and $y=0$ yields the intersection coordinates of the plane and the $y-$axes.

The substitution of $x=0$ and $y=0$ in $a x+b y+c z=d$ gives:

$$\begin{gathered} a\left(0\right)+b\left(0\right)+cz=d \\ cz=d \\ z=\frac{d}{c} \end{gathered}$$

Therefore, coordinates of intersection of the plane and the $z-$axes is $\left(0,0,\frac{d}{c}\right)$

To draw the plane, take the three coordinates as the triangle's vertices and draw the plane using the triangle.