Plane Equation Derivation
The problem asks us to find the value of \(3p^2\), where \(p\) is the perpendicular distance from the origin \((0, 0, 0)\) to a plane passing through three specific points: \(A(1, 0, 0)\), \(B(0, 1, 0)\), and \(C(0, 0, 1)\).
These points are easily recognizable as the points where the plane intersects the coordinate axes. They represent the x-intercept, y-intercept, and z-intercept, respectively.
We can use the intercept form of the equation of a plane, which is given by:
\(\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\)Here, \(a\), \(b\), and \(c\) are the intercepts on the x, y, and z axes. From the given points, we have \(a = 1\), \(b = 1\), and \(c = 1\).
Substituting these values into the intercept form, we get the equation of the plane:
\(\frac{x}{1} + \frac{y}{1} + \frac{z}{1} = 1\)This simplifies to:
\(x + y + z = 1\)To use the distance formula, we rewrite the plane equation in the general form \(Ax + By + Cz + D = 0\):
\(x + y + z - 1 = 0\)From this form, we identify the coefficients: \(A = 1\), \(B = 1\), \(C = 1\), and \(D = -1\).
Distance Calculation
The perpendicular distance (\(p\)) from a point \((x_0, y_0, z_0)\) to the plane \(Ax + By + Cz + D = 0\) is calculated using the formula:
\(p = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}\)In this problem, the point is the origin, so \((x_0, y_0, z_0) = (0, 0, 0)\). The plane is \(x + y + z - 1 = 0\), with \(A = 1\), \(B = 1\), \(C = 1\), and \(D = -1\).
Plugging these values into the distance formula:
\(p = \frac{|1(0) + 1(0) + 1(0) + (-1)|}{\sqrt{1^2 + 1^2 + 1^2}}\) \(p = \frac{|0 + 0 + 0 - 1|}{\sqrt{1 + 1 + 1}}\) \(p = \frac{|-1|}{\sqrt{3}}\) \(p = \frac{1}{\sqrt{3}}\)So, the perpendicular distance from the origin to the plane is \(\frac{1}{\sqrt{3}}\).
Final Calculation of \(3p^2\)
The question asks for the value of \(3p^2\). We have calculated \(p = \frac{1}{\sqrt{3}}\).
First, let's find \(p^2\):
\(p^2 = \left( \frac{1}{\sqrt{3}} \right)^2\) \(p^2 = \frac{1^2}{(\sqrt{3})^2}\) \(p^2 = \frac{1}{3}\)Now, we multiply this result by 3:
\(3p^2 = 3 \times p^2\) \(3p^2 = 3 \times \frac{1}{3}\) \(3p^2 = 1\)Therefore, the value of \(3p^2\) is 1.
Similar Questions
- What are the direction ratios of a line M parallel to the plane P ?
Important Questions from Equation in a Plane
- What are the direction ratios of a line M parallel to the plane P ?
