What is the resultantof two vectors 7 units and 8 units acting at an angle of `45^(@)` to each other ?

Explanation

To find the resultant of two vectors with magnitudes 7 units and 8 units acting at an angle of 45 degrees to each other, we can use the formula derived from the parallelogram law of vector addition.

### Step-by-Step Solution:

1. **Identify the Vectors and Angle**:
- Let vector P = 7 units
- Let vector Q = 8 units
- Angle θ between P and Q = 45 degrees

2. **Use the Resultant Formula**:
The formula for the resultant R of two vectors P and Q acting at an angle θ is given by:
\[ R = \sqrt{P^2 + Q^2 + 2PQ \cos(\theta)} \]

3. **Substitute the Values**:
- Substitute P = 7, Q = 8, and θ = 45 degrees into the formula:
\[ R = \sqrt{7^2 + 8^2 + 2 \cdot 7 \cdot 8 \cdot \cos(45^\circ)} \]

4. **Calculate Each Term**:
- Calculate \(7^2 = 49\)
- Calculate \(8^2 = 64\)
- Calculate \(\cos(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.7071\)
- Now calculate \(2 \cdot 7 \cdot 8 \cdot \cos(45^\circ) = 2 \cdot 7 \cdot 8 \cdot \frac{1}{\sqrt{2}} = 112 \cdot \frac{1}{\sqrt{2}} \approx 79.37\)

5. **Combine the Terms**:
\[ R = \sqrt{49 + 64 + 79.37} \]
\[ R = \sqrt{192.37} \]

6. **Calculate the Resultant**:
\[ R \approx 13.86 \text{ units} \]

7. **Determine the Angle with Respect to Vector P**:
To find the angle α that the resultant makes with vector P, we can use the tangent function:
\[ \tan(\alpha) = \frac{Q \sin(\theta)}{P + Q \cos(\theta)} \]
- Substitute the values:
\[ \tan(\alpha) = \frac{8 \sin(45^\circ)}{7 + 8 \cos(45^\circ)} \]
- Calculate \(\sin(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.7071\) and \(\cos(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.7071\):
\[ \tan(\alpha) = \frac{8 \cdot 0.7071}{7 + 8 \cdot 0.7071} \]
- Calculate the denominator:
\[ 7 + 8 \cdot 0.7071 \approx 7 + 5.6568 \approx 12.6568 \]
- Therefore,
\[ \tan(\alpha) \approx \frac{5.6568}{12.6568} \approx 0.447 \]
- Finally, calculate \(\alpha\):
\[ \alpha \approx \tan^{-1}(0.447) \approx 24.08^\circ \]

### Final Answer:
The resultant of the two vectors is approximately **13.86 units**, and it makes an angle of approximately **24.08 degrees** with vector P.