To find the value of \( \lambda \) in the expression \( \vec{a} \times (\vec{b} \times \vec{c}) = \vec{b} + \lambda \vec{c} \), we start with the given vectors:
Using the vector triple product identity, we have:
\[\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}\]
Plugging in the given expression, we need:
\[(\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} = \vec{b} + \lambda \vec{c}\]
From the equation above, we equate coefficients:
\[(\vec{a} \cdot \vec{c})\vec{b} = \vec{b}\]
\[-(\vec{a} \cdot \vec{b})\vec{c} = \lambda \vec{c}\]
This implies:
Next, calculate \(\vec{a} \cdot \vec{b}\):
\[\vec{a} \cdot \vec{b} = (3\hat{i} + \hat{j}) \cdot (\hat{i} + 2\hat{j} + \hat{k})\]
\[= 3 \times 1 + 1 \times 2 + 0 = 3 + 2 = 5\]
Thus, \( \vec{a} \cdot \vec{b} = 5 \). Plugging this into the equation:
\[-5 = \lambda\]
Therefore, the value of \( \lambda \) is \(-5\).
The correct answer is \(-5\).
\(\vec{a} = 3\hat{i} + \hat{j} \quad \text{and} \quad \vec{b} = \hat{i} + 2\hat{j} + \hat{k}\)
\(\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} = \vec{b} + \lambda \vec{c}\)
If \(\vec{b} \quad \text{and} \quad \vec{c}\) are non-parallel, then
\(\vec{a} \cdot \vec{c} = 1 \quad \text{and} \quad \vec{a} \cdot \vec{b} = -\lambda\)
but \(\vec{a} \cdot \vec{b} = 5\)
\(⇒λ=−5\)
So, the correct option is (A): -5
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?

In the given figure, the blocks $A$, $B$ and $C$ weigh $4\,\text{kg}$, $6\,\text{kg}$ and $8\,\text{kg}$ respectively. The coefficient of sliding friction between any two surfaces is $0.5$. The force $\vec{F}$ required to slide the block $C$ with constant speed is ___ N.
(Given: $g = 10\,\text{m s}^{-2}$) 
A vector is an object which has both magnitudes and direction. It is usually represented by an arrow which shows the direction(→) and its length shows the magnitude. The arrow which indicates the vector has an arrowhead and its opposite end is the tail. It is denoted as
The magnitude of the vector is represented as |V|. Two vectors are said to be equal if they have equal magnitudes and equal direction.
Arithmetic operations such as addition, subtraction, multiplication on vectors. However, in the case of multiplication, vectors have two terminologies, such as dot product and cross product.