Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2 \hat{i} + 2 \hat{j} + \hat{k}$ and $\vec{d} = \vec{a} \times \vec{b}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c} = |\vec{c}|$, $|\vec{c} - 2 \vec{a}|^2 = 8$ and the angle between $\vec{d}$ and $\vec{c}$ is $\frac{\pi}{4}$, then $|10 - 3 \vec{b} \cdot \vec{c}| + |\vec{d} \times \vec{c}|^2$ is equal to

Step 1: Calculate vectors $\vec{a}$, $\vec{b}$, and $\vec{d}$ and their magnitudes

First, we calculate the magnitudes of vectors $\vec{a}$ and $\vec{b}$. Then, we compute the cross product $\vec{d} = \vec{a} \times \vec{b}$ using the determinant formula. Finally, we find the magnitude of vector $\vec{d}$. These values will be used in subsequent calculations.

Step 2: Determine the magnitude of vector $\vec{c}$

We use the given condition $|\vec{c} - 2 \vec{a}|^2 = 8$. Expanding the dot product, we get $|\vec{c}|^2 - 4 \vec{a} \cdot \vec{c} + 4 |\vec{a}|^2 = 8$. Substituting the given $\vec{a} \cdot \vec{c} = |\vec{c}|$ and the calculated $|\vec{a}| = \sqrt{3}$, we form a quadratic equation in terms of $|\vec{c}|$. Solving this equation yields $|\vec{c}| = 2$.

Step 3: Calculate $\vec{b} \cdot \vec{c}$

We use the given conditions $\vec{a} \cdot \vec{c} = |\vec{c}|$ and the angle between $\vec{d}$ and $\vec{c}$ is $\frac{\pi}{4}$ to find $\vec{d} \cdot \vec{c}$. We express $\vec{c}$ as a linear combination of $\vec{a}$, $\vec{d}$, and a vector perpendicular to both, which is parallel to $\vec{a} \times \vec{d}$. By substituting the known dot products and magnitudes, we solve for the scalar coefficient $k$. This yields two possible values for $k$, leading to two possible vectors for $\vec{c}$ and consequently two possible values for $\vec{b} \cdot \vec{c}$.

Step 4: Calculate $|\vec{d} \times \vec{c}|^2$

We use the formula for the magnitude of the cross product: $|\vec{d} \times \vec{c}| = |\vec{d}||\vec{c}|\sin\theta$, where $\theta$ is the angle between $\vec{d}$ and $\vec{c}$. We substitute the calculated magnitudes of $\vec{d}$ and $\vec{c}$, and the given angle $\frac{\pi}{4}$, to find $|\vec{d} \times \vec{c}|^2$.

Step 5: Calculate the final expression for both cases

We substitute the two possible values for $\vec{b} \cdot \vec{c}$ and the calculated value of $|\vec{d} \times \vec{c}|^2$ into the given expression. In both cases, the result is 6.

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