Let the cosine of angle between the vectors p and q be `lamda` such that `2p+q=hati+hatj and p+2q=hati-hatj`, then `lamda` is equal to

Options

A. `(5)/(9)`
B. `-(4)/(5)`
C. `(3)/(9)`
D. `(7)/(9)`

Answer

To solve the problem, we need to find the cosine of the angle between the vectors \( \mathbf{p} \) and \( \mathbf{q} \) given the equations:

1. \( 2\mathbf{p} + \mathbf{q} = \hat{i} + \hat{j} \) (Equation 1)
2. \( \mathbf{p} + 2\mathbf{q} = \hat{i} - \hat{j} \) (Equation 2)

### Step 1: Multiply Equation 1 by 2
We start by multiplying Equation 1 by 2 to eliminate \( \mathbf{q} \):

\[ 4\mathbf{p} + 2\mathbf{q} = 2\hat{i} + 2\hat{j} \quad \text{(Equation 3)} \]

### Step 2: Write Equation 2
We will keep Equation 2 as it is:

\[ \mathbf{p} + 2\mathbf{q} = \hat{i} - \hat{j} \quad \text{(Equation 2)} \]

### Step 3: Subtract Equation 2 from Equation 3
Now, we subtract Equation 2 from Equation 3:

\[ (4\mathbf{p} + 2\mathbf{q}) - (\mathbf{p} + 2\mathbf{q}) = (2\hat{i} + 2\hat{j}) - (\hat{i} - \hat{j}) \]

This simplifies to:

\[ 3\mathbf{p} = \hat{i} + 3\hat{j} \]

### Step 4: Solve for \( \mathbf{p} \)
Now, we can solve for \( \mathbf{p} \):

\[ \mathbf{p} = \frac{1}{3}\hat{i} + \hat{j} \]

### Step 5: Substitute \( \mathbf{p} \) back into Equation 1
Now, we substitute \( \mathbf{p} \) back into Equation 1 to find \( \mathbf{q} \):

\[ 2\left(\frac{1}{3}\hat{i} + \hat{j}\right) + \mathbf{q} = \hat{i} + \hat{j} \]

This simplifies to:

\[ \frac{2}{3}\hat{i} + 2\hat{j} + \mathbf{q} = \hat{i} + \hat{j} \]

### Step 6: Solve for \( \mathbf{q} \)
Now, we can isolate \( \mathbf{q} \):

\[ \mathbf{q} = \hat{i} + \hat{j} - \left(\frac{2}{3}\hat{i} + 2\hat{j}\right) \]

This simplifies to:

\[ \mathbf{q} = \left(1 - \frac{2}{3}\right)\hat{i} + \left(1 - 2\right)\hat{j} = \frac{1}{3}\hat{i} - \hat{j} \]

### Step 7: Calculate \( \lambda \)
Now that we have both \( \mathbf{p} \) and \( \mathbf{q} \), we can find \( \lambda \) using the formula:

\[ \lambda = \frac{\mathbf{p} \cdot \mathbf{q}}{|\mathbf{p}| |\mathbf{q}|} \]

#### Step 7.1: Calculate \( \mathbf{p} \cdot \mathbf{q} \)
First, we calculate the dot product:

\[ \mathbf{p} \cdot \mathbf{q} = \left(\frac{1}{3}\hat{i} + \hat{j}\right) \cdot \left(\frac{1}{3}\hat{i} - \hat{j}\right) = \frac{1}{3} \cdot \frac{1}{3} + 1 \cdot (-1) = \frac{1}{9} - 1 = -\frac{8}{9} \]

#### Step 7.2: Calculate \( |\mathbf{p}| \) and \( |\mathbf{q}| \)
Now we calculate the magnitudes:

\[ |\mathbf{p}| = \sqrt{\left(\frac{1}{3}\right)^2 + 1^2} = \sqrt{\frac{1}{9} + 1} = \sqrt{\frac{10}{9}} = \frac{\sqrt{10}}{3} \]

\[ |\mathbf{q}| = \sqrt{\left(\frac{1}{3}\right)^2 + (-1)^2} = \sqrt{\frac{1}{9} + 1} = \sqrt{\frac{10}{9}} = \frac{\sqrt{10}}{3} \]

### Step 8: Substitute into \( \lambda \)
Now we substitute into the formula for \( \lambda \):

\[ \lambda = \frac{-\frac{8}{9}}{\left(\frac{\sqrt{10}}{3}\right) \left(\frac{\sqrt{10}}{3}\right)} = \frac{-\frac{8}{9}}{\frac{10}{9}} = -\frac{8}{10} = -\frac{4}{5} \]

### Final Answer
Thus, the value of \( \lambda \) is:

\[ \lambda = -\frac{4}{5} \]