Question

The angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{3}\). If \(\|\vec{a}\| = 5\) and \(\|\vec{b}\| = 10\), then \(\|\vec{a} + \vec{b}\|\) is equal to:

• A. \(7\sqrt{5}\)
• B. \(5\sqrt{5}\)
• C. 15
• D. \(5\sqrt{3}\)
• E. \(5\sqrt{7}\)

Hint:

When using the Law of Cosines to find the magnitude of the vector sum, ensure that the angle used is the one between the vectors, as this will significantly impact the result.

Answer 1

Answer: (E)

The magnitude of the vector sum \(\vec{a} + \vec{b}\) can be found using the Law of Cosines in vector form: \[ \|\vec{a} + \vec{b}\|^2 = \|\vec{a}\|^2 + \|\vec{b}\|^2 + 2\|\vec{a}\|\|\vec{b}\|\cos(\theta) \] Given \(\|\vec{a}\| = 5\), \(\|\vec{b}\| = 10\), and \(\theta = \frac{\pi}{3}\) (angle between the vectors): \[ \|\vec{a} + \vec{b}\|^2 = 5^2 + 10^2 + 2 \cdot 5 \cdot 10 \cdot \cos\left(\frac{\pi}{3}\right) \] \[ = 25 + 100 + 100 \cdot \frac{1}{2} \] \[ = 25 + 100 + 50 = 175 \] \[ \|\vec{a} + \vec{b}\| = \sqrt{175} = 5\sqrt{7} \] Thus, the magnitude of \(\vec{a} + \vec{b}\) is \(5\sqrt{7}\).