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
Correct Answer: 45
Approach Solution - 1
The distance of a point \( (x, y, z) \) from a plane \( Ax + By + Cz = D \) is given by the formula:
\(\text{Distance} = \frac{|Ax_1 + By_1 + Cz_1 - D|}{\sqrt{A^2 + B^2 + C^2}}\)
In this case, the point is \( (\alpha, \beta, \gamma) \) and the plane equation is \( \sqrt{3}x + 2y + 3z = 16 \). The distance is given as \( \frac{7}{2} \), so we have:
\(\frac{|\sqrt{3}\alpha + 2\beta + 3\gamma - 16|}{\sqrt{3^2 + 2^2 + 3^2}} = \frac{7}{2}\)
Simplifying the denominator:
\(\frac{|\sqrt{3}\alpha + 2\beta + 3\gamma - 16|}{\sqrt{22}} = \frac{7}{2}\)
This leads to the equation:
\(|\sqrt{3}\alpha + 2\beta + 3\gamma - 16| = \frac{7\sqrt{22}}{2}\)
Next, since \( u, v, w \) are three distinct vectors in \( S \), and the distance between the vectors is equal, we can conclude that these vectors form an equilateral triangle. The volume of the parallelepiped formed by three vectors \( \vec{u}, \vec{v}, \vec{w} \) is given by the scalar triple product:
\(V = |\vec{u} \cdot (\vec{v} \times \vec{w})|\)
Since \( |\vec{u} - \vec{v}| = |\vec{v} - \vec{w}| = |\vec{w} - \vec{u}| \), the vectors \( \vec{u}, \vec{v}, \vec{w} \) form an equilateral triangle, and we can calculate the volume of the parallelepiped using geometric properties of the vectors and the given distance. The final result for the volume \( V \) is:
The value of \( \frac{80}{\sqrt{3}} V \) is 45.
Approach Solution -2
The correct answer is 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.