We are given two unit vectors $\vec{a}$ and $\vec{b}$, inclined at an angle of $\theta = \frac{\pi}{3}$. We are tasked with simplifying the following expression:

$-\left[ \vec{a} \times \vec{b} + \left( \vec{a} \cdot \vec{b} \right) \vec{b} \right]$

Step-by-Step Breakdown:

1. Cross product ($\vec{a} \times \vec{b}$):

The magnitude of the cross product of two vectors $\vec{a}$ and $\vec{b}$ is given by: $|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin\theta$ Since $\vec{a}$ and $\vec{b}$ are unit vectors, $|\vec{a}| = |\vec{b}| = 1$, and $\theta = \frac{\pi}{3}$, we get: $|\vec{a} \times \vec{b}| = 1 \cdot 1 \cdot \sin\left(\frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$ Thus, the cross product $\vec{a} \times \vec{b}$ has a magnitude of $\frac{\sqrt{3}}{2}$ and is perpendicular to both $\vec{a}$ and $\vec{b}$.

2. Dot product ($\vec{a} \cdot \vec{b}$):

The dot product of two vectors is given by: $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta$ Again, using $|\vec{a}| = |\vec{b}| = 1$ and $\theta = \frac{\pi}{3}$, we get: $\vec{a} \cdot \vec{b} = 1 \cdot 1 \cdot \cos\left(\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$

3. Expression Simplification:

We now substitute these results into the original expression:

$-\left[ \vec{a} \times \vec{b} + \left( \vec{a} \cdot \vec{b} \right) \vec{b} \right]$ This becomes: $-\left[ \vec{a} \times \vec{b} + \frac{1}{2} \vec{b} \right]$

Since $\vec{a} \times \vec{b}$ is perpendicular to $\vec{b}$, we cannot directly combine these terms. Therefore, the simplified expression is:

$-\left( \vec{a} \times \vec{b} + \frac{1}{2} \vec{b} \right)$

Finally, distributing the negative sign:

$-\vec{a} \times \vec{b} - \frac{1}{2} \vec{b}$

This is the simplified form of the given vector expression.

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Would you like further details or have any questions?

Here are 5 related questions for practice:

1. What is the geometric interpretation of a cross product?
2. How do the cross product and dot product differ in vector operations?
3. What are the properties of unit vectors?
4. How does the angle between two vectors affect their cross product?
5. What is the result of a cross product when two vectors are parallel?

Tip: Always remember that the cross product of two vectors results in a vector perpendicular to the plane containing the two original vectors!