Question

If \( \mathbf{A} = 4\hat{i} + 3\hat{j} \) and \( \mathbf{B} = 3\hat{i} + 4\hat{j} \), then the cosine of the angle between \( \vec{A} \) and \( \vec{A} + \vec{B} \) is:

• A. \( \frac{9\sqrt{2}}{5} \)
• B. \( \frac{7}{5\sqrt{2}} \)
• C. \( \frac{5\sqrt{2}}{49} \)
• D. \( \frac{5\sqrt{2}}{28} \)

Hint:

To calculate the cosine of the angle between two vectors, use the formula \( \cos \theta = \frac{\vec{A} \cdot \vec{C}}{|\vec{A}| |\vec{C}|} \). Remember to calculate both the dot product and magnitudes of the vectors.

Answer 1

The Correct Option is B

Step 1: Cosine of the Angle Between Two Vectors
The cosine of the angle \( \theta \) between two vectors \( \vec{A} \) and \( \vec{C} = \vec{A} + \vec{B} \) is given by: \[ \cos \theta = \frac{\vec{A} \cdot \vec{C}}{|\vec{A}| |\vec{C}|} \] where \( \vec{A} \cdot \vec{C} \) is the dot product of the vectors and \( |\vec{A}| \) and \( |\vec{C}| \) are the magnitudes of the vectors.
Step 2: Calculate the Dot Product \( \vec{A} \cdot \vec{C} \)
First, calculate \( \vec{C} = \vec{A} + \vec{B} = (4\hat{i} + 3\hat{j}) + (3\hat{i} + 4\hat{j}) = 7\hat{i} + 7\hat{j} \).
The dot product \( \vec{A} \cdot \vec{C} \) is: \[ \vec{A} \cdot \vec{C} = (4\hat{i} + 3\hat{j}) \cdot (7\hat{i} + 7\hat{j}) = 4 \cdot 7 + 3 \cdot 7 = 28 + 21 = 49 \]
Step 3: Calculate the Magnitudes of \( \vec{A} \) and \( \vec{C} \)
The magnitude of \( \vec{A} \) is: \[ |\vec{A}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
The magnitude of \( \vec{C} \) is: \[ |\vec{C}| = \sqrt{7^2 + 7^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2} \]
Step 4: Calculate the Cosine of the Angle
Now, substitute the values into the cosine formula: \[ \cos \theta = \frac{49}{5 \times 7\sqrt{2}} = \frac{49}{35\sqrt{2}} = \frac{7}{5\sqrt{2}} \] Final Answer: The cosine of the angle between \( \vec{A} \) and \( \vec{A} + \vec{B} \) is \( \frac{7}{5\sqrt{2}} \).