Two vectors having magnitude 12 and 13 are inclined at and angle` 45^(@)` to each other.find their resultant vector.

Answer

To find the resultant vector of two vectors with magnitudes 12 and 13 inclined at an angle of 45 degrees to each other, we can follow these steps:

### Step 1: Identify the vectors and their magnitudes
Let vector A have a magnitude of 12 and vector B have a magnitude of 13. The angle (θ) between them is 45 degrees.

### Step 2: Use the formula for the magnitude of the resultant vector
The magnitude of the resultant vector (C) can be calculated using the formula:
\[ |C| = \sqrt{A^2 + B^2 + 2AB \cos(\theta)} \]
Substituting the values:
\[ |C| = \sqrt{12^2 + 13^2 + 2 \cdot 12 \cdot 13 \cdot \cos(45^\circ)} \]

### Step 3: Calculate the squares of the magnitudes
Calculate \(12^2\) and \(13^2\):
\[ 12^2 = 144 \]
\[ 13^2 = 169 \]

### Step 4: Calculate \(2AB \cos(\theta)\)
First, calculate \(2 \cdot 12 \cdot 13\):
\[ 2 \cdot 12 \cdot 13 = 312 \]
Now, calculate \(\cos(45^\circ)\):
\[ \cos(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.7071 \]
Now multiply:
\[ 312 \cdot \cos(45^\circ) \approx 312 \cdot 0.7071 \approx 220.88 \]

### Step 5: Substitute back into the formula
Now substitute these values back into the formula for the magnitude of C:
\[ |C| = \sqrt{144 + 169 + 220.88} \]
Calculating the sum:
\[ |C| = \sqrt{533.88} \]

### Step 6: Calculate the square root
Now, find the square root:
\[ |C| \approx 23.09 \]

### Step 7: Calculate the angle α with respect to vector B
To find the angle α that the resultant vector makes with vector B, we can use the formula:
\[ \alpha = \tan^{-1}\left(\frac{A \sin(\theta)}{B + A \cos(\theta)}\right) \]
Substituting the known values:
\[ \alpha = \tan^{-1}\left(\frac{12 \cdot \sin(45^\circ)}{13 + 12 \cdot \cos(45^\circ)}\right) \]

### Step 8: Calculate \(\sin(45^\circ)\) and \(\cos(45^\circ)\)
We already know:
\[ \sin(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.7071 \]
Substituting this value:
\[ \alpha = \tan^{-1}\left(\frac{12 \cdot 0.7071}{13 + 12 \cdot 0.7071}\right) \]

### Step 9: Calculate the numerator and denominator
Numerator:
\[ 12 \cdot 0.7071 \approx 8.49 \]
Denominator:
\[ 13 + 12 \cdot 0.7071 \approx 13 + 8.49 \approx 21.49 \]

### Step 10: Calculate α
Now substitute these into the formula:
\[ \alpha = \tan^{-1}\left(\frac{8.49}{21.49}\right) \]
Calculating the value:
\[ \alpha \approx \tan^{-1}(0.394) \approx 21.55^\circ \]

### Final Result
The magnitude of the resultant vector is approximately **23.09** and the angle with vector B is approximately **21.55 degrees**.

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