The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vector $\widehat{a},\widehat{b},\widehat{c}$ such that $\widehat{a}.\widehat{b}=\widehat{b}.\widehat{c}=\widehat{c}.\widehat{a}=\frac{1}{2}.$Then, the volume of the parallelopiped is
- $\frac{1}{\sqrt{2}}$
- $\frac{1}{2\sqrt{2}} $
- $\frac{\sqrt{3}}{2}$
- $\frac{1}{\sqrt{3}}$
The Correct Option is A
Solution and Explanation
as $\widehat{a},\widehat{b},\widehat{c}$ is given by $[\widehat{a},\widehat{b},\widehat{c}]=\widehat{a}.(\widehat{b}\times\widehat{c})$
$Now,\ [\widehat{a}.\widehat{b}\widehat{c}]^2=\begin{array}| \widehat{a}.\widehat{a}&\widehat{a}.\widehat{b}&\widehat{a}.\widehat{c}\\\widehat{b}.\widehat{a}&\widehat{b}.\widehat{b}&\widehat{b}.\widehat{c}\\\widehat{c}.\widehat{a}&\widehat{c}.\widehat{b}&\widehat{c}.\widehat{c}\\\end{array}=\begin{array}|1&1/2&1/2\\1/2&1&1/2\\1/2&1/2&1\\\end{array}$
$\Rightarrow \ \ \ \ \ \ [\widehat{a}.\widehat{b}\widehat{c}]^2= 1\Bigg(1-\frac{1}{4}\Bigg)-\frac{1}{2}\Bigg(\frac{1}{2}\frac{1}{2}\Bigg)+\frac{1}{2}\Bigg(\frac{1}{4}\frac{1}{2}\Bigg)=\frac{1}{2}$
Thus, the required volume of the parallelopiped
$=\frac{1}{\sqrt{2}} cu unit$
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Concepts Used:
Vectors
The quantities having magnitude as well as direction are known as Vectors or Vector quantities. Vectors are the objects which are found in accumulated form in vector spaces accompanying two types of operations. These operations within the vector space include the addition of two vectors and multiplication of the vector with a scalar quantity. These operations can alter the proportions and order of the vector but the result still remains in the vector space. It is often recognized by symbols such as U ,V, and W
Representation of a Vector :
A line having an arrowhead is known as a directed line. A segment of the directed line has both direction and magnitude. This segment of the directed line is known as a vector. It is represented by a or commonly as AB. In this line segment AB, A is the starting point and B is the terminal point of the line.
Types of Vectors:
Here we will be discussing different types of vectors. There are commonly 10 different types of vectors frequently used in maths. The 10 types of vectors are:
- Zero vector
- Unit Vector
- Position Vector
- Co-initial Vector
- Like and Unlike Vectors
- Coplanar Vector
- Collinear Vector
- Equal Vector
- Displacement Vector
- Negative of a Vector