Q 6. - Calculus

Question

Explain why there are infinitely many different planes containing any given line in ${\mathrm{ℝ}}^3$ What form does the equation of a plane containing the $x-$axis have?

Step-by-Step Solution

Answer

The form of the plane containing the $x-$axes is of the type $a x=d$ with $a$ and $d$ being nonzero constants.

Step 1: Given information

The planes in ${\mathrm{ℝ}}^3$

Step 2: Calculation

The goal is to explain why there are an endless number of planes in ${\mathrm{ℝ}}^3$ that include any line.

The plane in ${\mathrm{ℝ}}^3$ is defined by a point $(a, b, c)$ on the plane and two direction vectors parallel to the plane.

The direction vectors come in a variety of shapes and sizes. That's why each given line in ${\mathrm{ℝ}}^3$ can be found on an endless number of different planes.

The equation of the plane containing the $x-$axes have $y$ and $z$ coordinates equal to $0$

Thus, the form of the plane containing the $x-$axes is of the type $a x=d$ with $a$ and $d$ being nonzero constants.