Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a} + 2 \vec{b}$ and $3 \vec{a} - \lambda \vec{b}$ are perpendicular to each other, then the number of values of $\lambda$ in $[-1{,}3]$ is:

Step 1: Understand Given Information

We are given that $\vec{a}$ and $\vec{b}$ are unit vectors, which means their magnitudes are 1. The angle between them is given as $\frac{\pi}{3}$ radians. This information will be used to calculate their dot product.

Step 2: Calculate Dot Product of Vectors a and b

The dot product of two vectors is defined as the product of their magnitudes and the cosine of the angle between them. Substituting the given values, we find that the dot product of $\vec{a}$ and $\vec{b}$ is $\frac{1}{2}$.

Step 3: Apply Perpendicularity Condition

Two vectors are perpendicular if and only if their dot product is zero. We set the dot product of the two given vectors, $(\lambda \vec{a} + 2 \vec{b})$ and $(3 \vec{a} - \lambda \vec{b})$, equal to zero.

Step 4: Expand and Solve the Equation for Lambda

We expand the dot product using the distributive property. We know that $\vec{a} \cdot \vec{a} = |\vec{a}|^2$ and $\vec{b} \cdot \vec{b} = |\vec{b}|^2$. Substituting the magnitudes and the dot product $\vec{a} \cdot \vec{b} = \frac{1}{2}$, we simplify the equation to a quadratic equation in terms of $\lambda$.

Step 5: Solve the Quadratic Equation for Lambda

We use the quadratic formula $\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to solve for $\lambda$. The two values obtained are $1 + \sqrt{7}$ and $1 - \sqrt{7}$.

Step 6: Check Values of Lambda in the Given Interval

We approximate the values of $\lambda$ to check if they fall within the interval $[-1, 3]$. We find that $1 + \sqrt{7} \approx 3.646$ and $1 - \sqrt{7} \approx -1.646$. Neither of these values lies within the interval $[-1, 3]$.

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