Question

Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors, and \(\theta\) be the angle between them. If \(\vec{a}-\vec{b}\) is a unit vector, then \(\theta\) is equal to

• A. \(\frac{\pi}{3}\)
• B. \(\frac{\pi}{2}\)
• C. \(\frac{\pi}{4}\)
• D. \(\frac{2\pi}{3}\)
• E. \(\frac{\pi}{6}\)

Hint:

For vector problems, always use identities like \(|\vec{a}-\vec{b}|^2=|\vec{a}|^2+|\vec{b}|^2-2\vec{a}\cdot\vec{b}\). It simplifies calculations quickly.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We are given information about the magnitudes of three vectors: \(\vec{a}\), \(\vec{b}\), and their difference \(\vec{a} - \vec{b}\). We need to find the angle \(\theta\) between \(\vec{a}\) and \(\vec{b}\). The magnitude of a sum or difference of vectors is related to the dot product and the angle between them.
Step 2: Key Formula or Approach:
The magnitude squared of a vector is the dot product of the vector with itself: \(|\vec{v}|^2 = \vec{v} \cdot \vec{v}\).
We will use this property for the vector \(\vec{a} - \vec{b}\).
\[ |\vec{a} - \vec{b}|^2 = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b}) \] Expanding the dot product:
\[ |\vec{a} - \vec{b}|^2 = \vec{a}\cdot\vec{a} - \vec{a}\cdot\vec{b} - \vec{b}\cdot\vec{a} + \vec{b}\cdot\vec{b} \] \[ |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 - 2(\vec{a}\cdot\vec{b}) + |\vec{b}|^2 \] We also use the definition of the dot product: \(\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta\).
Step 3: Detailed Explanation:
We are given the following information:
\(\vec{a}\) and \(\vec{b}\) are unit vectors, so \(|\vec{a}| = 1\) and \(|\vec{b}| = 1\).
\(\vec{a} - \vec{b}\) is a unit vector, so \(|\vec{a} - \vec{b}| = 1\).
Let's use the formula from Step 2:
\[ |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2(\vec{a}\cdot\vec{b}) \] Substitute the definition of the dot product:
\[ |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2|\vec{a}||\vec{b}|\cos\theta \] Now, plug in the given magnitudes:
\[ (1)^2 = (1)^2 + (1)^2 - 2(1)(1)\cos\theta \] \[ 1 = 1 + 1 - 2\cos\theta \] \[ 1 = 2 - 2\cos\theta \] \[ 2\cos\theta = 2 - 1 \] \[ 2\cos\theta = 1 \] \[ \cos\theta = \frac{1}{2} \] The angle \(\theta\) in the range \([0, \pi]\) for which \(\cos\theta = 1/2\) is \(\theta = \frac{\pi}{3}\).
Step 4: Final Answer:
The angle \(\theta\) between the vectors is \(\frac{\pi}{3}\). This corresponds to option (A).