`veca` has components 3P and 1 in rectangular cartesian system. `veca` is rotated counterclockwise about origin such that its components now become `sqrt10 and P+1` then a value of "P" is

Options

A. `-5/4`
B. `4/5`
C. 1
D. -1

Answer

To solve the problem, we need to find the value of \( P \) given that the vector \( \vec{a} \) has components \( 3P \) and \( 1 \) in a rectangular Cartesian system, and after a counterclockwise rotation about the origin, its components become \( \sqrt{10} \) and \( P + 1 \).

### Step-by-step Solution:

1. **Identify the initial components of the vector**:
The initial components of the vector \( \vec{a} \) are given as:
\[ \vec{a} = (3P, 1) \]

2. **Calculate the magnitude of the initial vector**:
The magnitude \( |\vec{a}| \) of the vector can be calculated using the formula:
\[ |\vec{a}| = \sqrt{(3P)^2 + (1)^2} = \sqrt{9P^2 + 1} \]

3. **Identify the components after rotation**:
After the counterclockwise rotation, the new components of the vector are:
\[ \vec{a'} = (\sqrt{10}, P + 1) \]

4. **Calculate the magnitude of the rotated vector**:
The magnitude of the rotated vector \( \vec{a'} \) is:
\[ |\vec{a'}| = \sqrt{(\sqrt{10})^2 + (P + 1)^2} = \sqrt{10 + (P + 1)^2} \]

5. **Set the magnitudes equal**:
Since the rotation does not change the magnitude of the vector, we equate the two magnitudes:
\[ \sqrt{9P^2 + 1} = \sqrt{10 + (P + 1)^2} \]

6. **Square both sides to eliminate the square roots**:
Squaring both sides gives:
\[ 9P^2 + 1 = 10 + (P + 1)^2 \]

7. **Expand the right-hand side**:
Expanding \( (P + 1)^2 \):
\[ (P + 1)^2 = P^2 + 2P + 1 \]
Therefore, the equation becomes:
\[ 9P^2 + 1 = 10 + P^2 + 2P + 1 \]

8. **Simplify the equation**:
Rearranging gives:
\[ 9P^2 + 1 = P^2 + 2P + 11 \]
\[ 9P^2 - P^2 - 2P - 11 + 1 = 0 \]
\[ 8P^2 - 2P - 10 = 0 \]

9. **Divide the entire equation by 2**:
To simplify, we divide by 2:
\[ 4P^2 - P - 5 = 0 \]

10. **Use the quadratic formula to find \( P \)**:
The quadratic formula is given by:
\[ P = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 4 \), \( b = -1 \), and \( c = -5 \). Thus,
\[ P = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 4 \cdot (-5)}}{2 \cdot 4} \]
\[ P = \frac{1 \pm \sqrt{1 + 80}}{8} \]
\[ P = \frac{1 \pm \sqrt{81}}{8} \]
\[ P = \frac{1 \pm 9}{8} \]

11. **Calculate the possible values of \( P \)**:
This gives us two possible solutions:
\[ P = \frac{10}{8} = \frac{5}{4} \quad \text{and} \quad P = \frac{-8}{8} = -1 \]

12. **Conclusion**:
The possible values for \( P \) are \( \frac{5}{4} \) and \( -1 \). Since the problem asks for the value of \( P \), and based on the options given, the answer is:
\[ P = -1 \]