A unit vector in the xy-plane that makes an angle of `pi/4` with the vector `hati+hatj` and an angle of 'pi/3' with the vector `3hati-4hatj` is

Options

A. `hati`
B. `(1)/(sqrt(2))(hati+hatj)`
C. `(1)/(sqrt(3))(hati+hatj+hatk)`
D. Not in existence

Answer

To find the unit vector in the xy-plane that makes an angle of \(\frac{\pi}{4}\) with the vector \(\hat{i} + \hat{j}\) and an angle of \(\frac{\pi}{3}\) with the vector \(3\hat{i} - 4\hat{j}\), we can follow these steps:

### Step 1: Define the unit vector
Let the unit vector be represented as:
\[ \vec{a} = x\hat{i} + y\hat{j} \]
Since it is a unit vector, we have:
\[ \sqrt{x^2 + y^2} = 1 \quad \text{(1)} \]

### Step 2: Use the angle condition with \(\hat{i} + \hat{j}\)
We know that \(\vec{a}\) makes an angle of \(\frac{\pi}{4}\) with \(\hat{i} + \hat{j}\). Using the cosine formula:
\[ \cos\left(\frac{\pi}{4}\right) = \frac{\vec{a} \cdot (\hat{i} + \hat{j})}{|\vec{a}| |(\hat{i} + \hat{j})|} \]
Substituting the values:
\[ \frac{1}{\sqrt{2}} = \frac{(x\hat{i} + y\hat{j}) \cdot (\hat{i} + \hat{j})}{1 \cdot \sqrt{2}} \]
Calculating the dot product:
\[ \vec{a} \cdot (\hat{i} + \hat{j}) = x + y \]
Thus, we have:
\[ \frac{1}{\sqrt{2}} = \frac{x + y}{\sqrt{2}} \]
This simplifies to:
\[ x + y = 1 \quad \text{(2)} \]

### Step 3: Use the angle condition with \(3\hat{i} - 4\hat{j}\)
Now, we use the angle condition with the vector \(3\hat{i} - 4\hat{j}\):
\[ \cos\left(\frac{\pi}{3}\right) = \frac{\vec{a} \cdot (3\hat{i} - 4\hat{j})}{|\vec{a}| |(3\hat{i} - 4\hat{j})|} \]
Substituting the values:
\[ \frac{1}{2} = \frac{(x\hat{i} + y\hat{j}) \cdot (3\hat{i} - 4\hat{j})}{1 \cdot 5} \]
Calculating the dot product:
\[ \vec{a} \cdot (3\hat{i} - 4\hat{j}) = 3x - 4y \]
Thus, we have:
\[ \frac{1}{2} = \frac{3x - 4y}{5} \]
Multiplying both sides by 5:
\[ \frac{5}{2} = 3x - 4y \quad \text{(3)} \]

### Step 4: Solve the equations
Now, we have two equations:
1. \(x + y = 1\) (Equation 2)
2. \(3x - 4y = \frac{5}{2}\) (Equation 3)

From Equation (2), we can express \(y\) in terms of \(x\):
\[ y = 1 - x \]
Substituting this into Equation (3):
\[ 3x - 4(1 - x) = \frac{5}{2} \]
Simplifying:
\[ 3x - 4 + 4x = \frac{5}{2} \]
\[ 7x - 4 = \frac{5}{2} \]
Adding 4 to both sides:
\[ 7x = \frac{5}{2} + 4 = \frac{5}{2} + \frac{8}{2} = \frac{13}{2} \]
Dividing by 7:
\[ x = \frac{13}{14} \]
Now substituting back to find \(y\):
\[ y = 1 - \frac{13}{14} = \frac{1}{14} \]

### Step 5: Write the final unit vector
Thus, the unit vector \(\vec{a}\) is:
\[ \vec{a} = \frac{13}{14}\hat{i} + \frac{1}{14}\hat{j} \]

### Final Answer
The required unit vector is:
\[ \frac{13}{14}\hat{i} + \frac{1}{14}\hat{j} \]