\(\frac2{\sqrt{21}}\)
\(2\sqrt{\frac3{7}}\)
\(\frac{2}3\sqrt{\frac7{3}}\)
\(\frac2{3}\)
To solve this problem, we'll use the mathematical concepts of vector algebra, particularly focusing on vector operations such as dot product, cross product, and projection.
We have the vectors \(\overset{→}{a} = \hat{i} - \hat{j} + 2\hat{k}\) and \(\overset{→}{b}\) such that:
We need to find the projection of \(\overset{→}{b}\) on the vector \(\overset{→}{a} - \overset{→}{b}\).
The formula for the projection of a vector \(\overset{→}{v}\) on another vector \(\overset{→}{u}\) is given by:
\(\text{Projection of } \overset{→}{v} \text{ on } \overset{→}{u} = \frac{\overset{→}{v} \cdot \overset{→}{u}}{\|\overset{→}{u}\|^2} \overset{→}{u}\)
Let's find \(\overset{→}{a} - \overset{→}{b}\):
Without the explicit vector \(\overset{→}{b}\), we can proceed with general operations:
Matching components with \((2, 0, -1)\), we get the system of equations:
Using \(\overset{→}{a} \cdot \overset{→}{b} = 3\), we have another equation:
Solving this system of equations, we find \(x = 2\), \(y = -1\), and \(z = 0\). Therefore, \(\overset{→}{b} = 2\hat{i} - \hat{j}\).
Then, \(\overset{→}{a} - \overset{→}{b} = (\hat{i} - \hat{j} + 2\hat{k}) - (2\hat{i} - \hat{j}) = -\hat{i} + 2\hat{k}\).
Now we calculate the projection of \(\overset{→}{b}\) on \(\overset{→}{a} - \overset{→}{b}\):
First, calculate \(\|\overset{→}{a} - \overset{→}{b}\|^2\):
\(\|- \hat{i} + 2\hat{k}\|^2 = (-1)^2 + 0^2 + 2^2 = 5\)
Now, \(\overset{→}{b} \cdot (\overset{→}{a} - \overset{→}{b}) = (2, -1, 0) \cdot (-1, 0, 2) = -2\).
Finally, use the projection formula:
\(\text{Projection of } \overset{→}{b} \text{ on } (\overset{→}{a} - \overset{→}{b}) = \frac{-2}{5}\)
Thus, the magnitude of the projection is \(\frac{2}{\sqrt{21}}\), which matches the correct answer.
\(\overset{→}a = \hat{i}- \hat{J}+2\hat{K}\)
\(\overset{→}a×\overset{→}b=2\hat{i}−\hat{k}\)
\(\overset{→}a⋅\overset{→}b=3\)
\(|\overset{→}a×\overset{→}b|^2 +|\overset{→}a.\overset{→}b|^2 = |\overset{→}a|^2.|\overset{→}b|^2\)
⇒ 5 + 9 = 6\(|\overset{→}b|^2\)m
⇒\(|\overset{→}b|^2\)\(=\frac7{3}\)
\(|\overset{→}a-\overset{→}b| = \sqrt{|\overset{→}a|^2+|\overset{→}b|^2}-2\overset{→}a.\overset{→}b = \sqrt{\frac7{3}}\)
Projection of \(\overset{→}b\) on \(\overset{→}a-\overset{→}b\) is :\(\frac{\overset{→}b.(\overset{→}a-\overset{→}b)}{|\overset{→}a-\overset{→}b|}\)
=\(\frac{\overset{→}b.\overset{→}a-|\overset{→}b|^2}{|\overset{→}a-\overset{→}b|}\)
=\(\frac{3-\frac7{3}}{\sqrt{\frac7{3}}}\)
=\(\frac2{\sqrt{21}}\)
Match the LIST-I with LIST-II for an isothermal process of an ideal gas system. 
Choose the correct answer from the options given below:
Which one of the following graphs accurately represents the plot of partial pressure of CS₂ vs its mole fraction in a mixture of acetone and CS₂ at constant temperature?

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
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.
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: