Question

Let \(|\vec{a}| = 6\) and \(|\vec{b}| = 10\). If \(\vec{a}\) and \(\vec{b}\) make angles \(25^\circ\) and \(85^\circ\), respectively, with the x-axis, then the value of \(|\vec{a} + \vec{b}|\) is equal to

• A. 14
• B. 12
• C. 13
• D. 11
• E. 15

Hint:

Always visualize vectors as lines from the origin. The "angle with the x-axis" is just a polar coordinate. The "angle between them" is what matters for the addition formula.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The sum of two vectors depends on their magnitudes and the angle between them. The angle between them is the difference of their individual angles with the x-axis.

Step 2: Key Formula or Approach:
\(|\vec{a} + \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}| \cos \theta}\).

Step 3: Detailed Explanation:

Step 1: Calculate the angle \(\theta\) between the two vectors.
\[ \theta = 85^\circ - 25^\circ = 60^\circ \]

Step 2: Substitute magnitudes and angle into the resultant formula.
\[ |\vec{a} + \vec{b}|^2 = 6^2 + 10^2 + 2(6)(10) \cos 60^\circ \]
\[ |\vec{a} + \vec{b}|^2 = 36 + 100 + 120 \left( \frac{1}{2} \right) \]
\[ |\vec{a} + \vec{b}|^2 = 136 + 60 = 196 \]
\[ |\vec{a} + \vec{b}| = \sqrt{196} = 14 \]

Step 4: Final Answer:
The value of \(|\vec{a} + \vec{b}|\) is 14.