The volume of the tetrahedron with vertices P (-1, 2, 0), Q ( 2, 1, -3), R (1, 0, 1) and S (3, -2, 3) is
- $\frac{1}{3}$
- $\frac{2}{3}$
- $\frac{1}{4}$
- $\frac{3}{4}$
The Correct Option is B
Solution and Explanation
Given vertices of the tetrahedron:
\(P(-1,2,0), \; Q(2,1,-3), \; R(1,0,1), \; S(3,-2,3)\)
Volume of tetrahedron is:
\(\displaystyle V = \frac{1}{6} \left| [\vec{PQ} \; \vec{PR} \; \vec{PS}] \right|\)
Compute vectors:
\(\vec{PQ} = (2+1)\hat{i} + (1-2)\hat{j} + (-3-0)\hat{k} = 3\hat{i} - \hat{j} - 3\hat{k}\)
\(\vec{PR} = (1+1)\hat{i} + (0-2)\hat{j} + (1-0)\hat{k} = 2\hat{i} - 2\hat{j} + \hat{k}\)
\(\vec{PS} = (3+1)\hat{i} + (-2-2)\hat{j} + (3-0)\hat{k} = 4\hat{i} - 4\hat{j} + 3\hat{k}\)
Now, compute the determinant:
\(\displaystyle \begin{vmatrix} 3 & -1 & -3\\ 2 & -2 & 1\\ 4 & -4 & 3 \end{vmatrix} = 4 \)
Thus,
\(\displaystyle V = \frac{1}{6} \times 4 = \frac{2}{3}\)
Therefore, the volume of the tetrahedron is \(\frac{2}{3}\) cubic units.
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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.