The vector `vec(OA)` where O is origin is given by `vec(OA) = 2 hati + 2 hatj` . Now it is rotated by `45^(@)` anticlockwise about O . What will be the new vector . ?

Options

A. `2 sqrt(2) hatj`
B. `2 hatj`
C. `2 hatj`
D. `2sqrt(2) hatj`

Answer

To solve the problem of finding the new vector after rotating the vector \(\vec{OA} = 2 \hat{i} + 2 \hat{j}\) by \(45^\circ\) anticlockwise about the origin \(O\), we can follow these steps:

### Step 1: Understand the Initial Vector
The initial vector \(\vec{OA}\) can be represented in Cartesian coordinates as:
\[ \vec{OA} = 2 \hat{i} + 2 \hat{j} \]
This means that the vector has an x-component of 2 and a y-component of 2.

**Hint:** Identify the components of the vector in the Cartesian coordinate system.

### Step 2: Determine the Angle of the Vector
To find the angle \(\theta\) that the vector makes with the positive x-axis, we can use the tangent function:
\[ \tan(\theta) = \frac{y}{x} = \frac{2}{2} = 1 \]
Thus, \(\theta = 45^\circ\).

**Hint:** Use the relationship \(\tan(\theta) = \frac{y}{x}\) to find the angle of the vector.

### Step 3: Apply the Rotation Formula
When rotating a vector \(\vec{r} = x \hat{i} + y \hat{j}\) by an angle \(\phi\) anticlockwise, the new coordinates \((x', y')\) can be calculated using the rotation formulas:
\[ x' = x \cos(\phi) - y \sin(\phi) \]
\[ y' = x \sin(\phi) + y \cos(\phi) \]
For our case, \(\phi = 45^\circ\), \(x = 2\), and \(y = 2\).

**Hint:** Use the rotation formulas to find the new coordinates after rotation.

### Step 4: Calculate the New Components
Substituting the values into the rotation formulas:
\[ x' = 2 \cos(45^\circ) - 2 \sin(45^\circ) = 2 \left(\frac{\sqrt{2}}{2}\right) - 2 \left(\frac{\sqrt{2}}{2}\right) = 0 \]
\[ y' = 2 \sin(45^\circ) + 2 \cos(45^\circ) = 2 \left(\frac{\sqrt{2}}{2}\right) + 2 \left(\frac{\sqrt{2}}{2}\right) = 2\sqrt{2} \]

**Hint:** Substitute the values of \(\cos(45^\circ)\) and \(\sin(45^\circ)\) to compute the new components.

### Step 5: Write the New Vector
The new vector after rotation is:
\[ \vec{OA'} = 0 \hat{i} + 2\sqrt{2} \hat{j} \]
This can be simplified to:
\[ \vec{OA'} = 2\sqrt{2} \hat{j} \]

**Hint:** Combine the new components to express the new vector in standard form.

### Conclusion
The new vector after rotating \(\vec{OA}\) by \(45^\circ\) anticlockwise about the origin is:
\[ \vec{OA'} = 2\sqrt{2} \hat{j} \]