Question

The magnitude of component of vector A = $3\hat{i}+4\hat{j}$ along vector $2\hat{i}+\hat{j}$ is $\alpha$. Then find the value of $\frac{\alpha^2}{4}$.

Answer 1

5

Answer 2

Let $\mathbf{A} = 3\hat{i} + 4\hat{j}$ and $\mathbf{B} = 2\hat{i} + \hat{j}$.
The magnitude of the component of vector $\mathbf{A}$ along vector $\mathbf{B}$ is given by the formula:
$\alpha = \frac{|\mathbf{A} \cdot \mathbf{B}|}{|\mathbf{B}|}$.

First, calculate the dot product of $\mathbf{A}$ and $\mathbf{B}$:
$\mathbf{A} \cdot \mathbf{B} = (3\hat{i} + 4\hat{j}) \cdot (2\hat{i} + \hat{j}) = (3)(2) + (4)(1) = 6 + 4 = 10$.

Next, calculate the magnitude of vector $\mathbf{B}$:
$|\mathbf{B}| = |2\hat{i} + \hat{j}| = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}$.

Now, calculate the magnitude of the component of $\mathbf{A}$ along $\mathbf{B}$, which is $\alpha$:
$\alpha = \frac{|\mathbf{A} \cdot \mathbf{B}|}{|\mathbf{B}|} = \frac{|10|}{\sqrt{5}} = \frac{10}{\sqrt{5}}$.

We are asked to find the value of $\frac{\alpha^2}{4}$.
First, calculate $\alpha^2$:
$\alpha^2 = \left(\frac{10}{\sqrt{5}}\right)^2 = \frac{10^2}{(\sqrt{5})^2} = \frac{100}{5} = 20$.

Now, calculate $\frac{\alpha^2}{4}$:
$\frac{\alpha^2}{4} = \frac{20}{4} = 5$.