Let P be the plane √3x+2y+3z =16 and let and let S = {\(\alpha \hat{i}+\beta \hat{j}+\gamma\hat{k}:\alpha^2+\beta^2+\gamma^2=1\) and the distance of (α, β, γ) from the plane P is \(\frac{7}{2}\) }. Let u, v, and w be three distinct vectors in s such that |\(\vec{u}-\vec{v}\)| = |\(\vec{v}-\vec{w}\)| = |\(\vec{w}-\vec{u}\)|. Let V be the volume of the parallelepiped determined by vectors \(\vec{u},\vec{v},\vec{w}\). Then the value of \(\frac{80}{\sqrt3}\)V is
Approach Solution - 1
Volume of Parallelepiped Determined by Vectors
Let \( P \) be the plane defined by the equation:
\[ \sqrt{3}x + 2y + 3z = 16 \]
Let \( S \) be the set of vectors \( \mathbf{S} = \{\alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} : \alpha^2 + \beta^2 + \gamma^2 = 1 \} \) and the distance of \( (\alpha, \beta, \gamma) \) from the plane \( P \) is given as \( \frac{7}{2} \).
Let \( u, v, \) and \( w \) be three distinct vectors in \( S \) such that:
\[ | \mathbf{u} - \mathbf{v} | = | \mathbf{v} - \mathbf{w} | = | \mathbf{w} - \mathbf{u} | \]
The quantity \( V \) represents the volume of the parallelepiped determined by the vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \). The value of \( 80V \) is given as:
\[ \boxed{45} \]
Approach Solution -2
The correct answer will be 45 
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Concepts Used:
Plane
A surface comprising all the straight lines that join any two points lying on it is called a plane in geometry. A plane is defined through any of the following uniquely:
- Using three non-collinear points
- Using a point and a line not on that line
- Using two distinct intersecting lines
- Using two separate parallel lines
Properties of a Plane:
- In a three-dimensional space, if there are two different planes than they are either parallel to each other or intersecting in a line.
- A line could be parallel to a plane, intersects the plane at a single point or is existing in the plane.
- If there are two different lines that are perpendicular to the same plane then they must be parallel to each other.
- If there are two separate planes which are perpendicular to the same line then they must be parallel to each other.