Let $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$, $\vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k}$ and a vector $\vec{c}$ be such that $(\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k}$ and $\vec{a} \cdot \vec{c} = 3$. If $\vec{b} \times \vec{c} = \vec{d}$, then $|\vec{a} \cdot \vec{d}|$ is equal to:

Step 1: Expand the cross product

We begin by expanding the given cross product using the distributive property of the cross product. This allows us to separate the terms involving $\vec{a}$ and $\vec{c}$.

Step 2: Substitute and rearrange the equation

We substitute the expanded form back into the given equation. We are also given that $\vec{b} \times \vec{c} = \vec{d}$. Since the cross product is anti-commutative, $\vec{c} \times \vec{b} = -(\vec{b} \times \vec{c}) = -\vec{d}$. Substituting this into the equation simplifies it further.

Step 3: Calculate $\vec{a} \times \vec{b}$

Now, we calculate the cross product of $\vec{a}$ and $\vec{b}$ using the determinant formula. This gives us a specific vector value for $\vec{a} \times \vec{b}$.

Step 4: Solve for $\vec{d}$

Substitute the calculated value of $\vec{a} \times \vec{b}$ into the equation from step 2. Then, rearrange the equation to solve for $\vec{d}$.

Step 5: Calculate $\vec{a} \cdot \vec{d}$

Finally, we calculate the dot product of $\vec{a}$ and $\vec{d}$ using their component forms. The dot product is the sum of the products of corresponding components.

Step 6: Find the magnitude of $\vec{a} \cdot \vec{d}$

The problem asks for the magnitude of $\vec{a} \cdot \vec{d}$, which is the absolute value of the scalar result obtained in the previous step.

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