Given vectors $\mathbf{A} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\mathbf{B} = -\hat{i} + \hat{j} + 4\hat{k}$, and $\mathbf{C} = 3\hat{i} - 3\hat{j} - 12\hat{k}$, we need to find the angle between the vector $\mathbf{A} + \mathbf{B} + \mathbf{C}$ and the cross product $\mathbf{A} \times \mathbf{B}$.

1. Find $\mathbf{A} + \mathbf{B} + \mathbf{C}$:

$\mathbf{A} + \mathbf{B} + \mathbf{C} = (\hat{i} + 2\hat{j} + 3\hat{k}) + (-\hat{i} + \hat{j} + 4\hat{k}) + (3\hat{i} - 3\hat{j} - 12\hat{k})$

$= (\hat{i} - \hat{i} + 3\hat{i}) + (2\hat{j} + \hat{j} - 3\hat{j}) + (3\hat{k} + 4\hat{k} - 12\hat{k})$

$= 3\hat{i} + 0\hat{j} - 5\hat{k}$

So, $\mathbf{A} + \mathbf{B} + \mathbf{C} = 3\hat{i} - 5\hat{k}$.

2. Find the cross product $\mathbf{A} \times \mathbf{B}$:

\hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ -1 & 1 & 4 \end{vmatrix}$$ $$= \hat{i} \left(2 \cdot 4 - 3 \cdot 1\right) - \hat{j} \left(1 \cdot 4 - 3 \cdot (-1)\right) + \hat{k} \left(1 \cdot 1 - 2 \cdot (-1)\right)$$ $$= \hat{i} (8 - 3) - \hat{j} (4 + 3) + \hat{k} (1 + 2)$$ $$= 5\hat{i} - 7\hat{j} + 3\hat{k}$$ So, $$\mathbf{A} \times \mathbf{B} = 5\hat{i} - 7\hat{j} + 3\hat{k}$$. 3. **Find the angle between $$\mathbf{A} + \mathbf{B} + \mathbf{C}$$ and $$\mathbf{A} \times \mathbf{B}$$:** Let $$\mathbf{D} = \mathbf{A} + \mathbf{B} + \mathbf{C}$$ and $$\mathbf{E} = \mathbf{A} \times \mathbf{B}$$. $$\mathbf{D} = 3\hat{i} - 5\hat{k}$$ $$\mathbf{E} = 5\hat{i} - 7\hat{j} + 3\hat{k}$$ The angle $$\theta$$ between $$\mathbf{D}$$ and $$\mathbf{E}$$ is given by: $$\cos \theta = \frac{\mathbf{D} \cdot \mathbf{E}}{|\mathbf{D}| |\mathbf{E}|}$$ First, calculate $$\mathbf{D} \cdot \mathbf{E}$$: $$\mathbf{D} \cdot \mathbf{E} = (3\hat{i} - 5\hat{k}) \cdot (5\hat{i} - 7\hat{j} + 3\hat{k})$$ $$= 3 \cdot 5 + 0 \cdot (-7) + (-5) \cdot 3$$ $$= 15 - 15$$ $$= 0$$ Since the dot product is zero, the vectors are orthogonal, and thus the angle between them is $$90^\circ$$. **Summary:** The angle between the vector $$\mathbf{A} + \mathbf{B} + \mathbf{C}$$ and the cross product $$\mathbf{A} \times \mathbf{B}$$ is $$90^\circ$$. Would you like any further details or have any questions? Here are 8 related questions to explore further: 1. How do you compute the magnitude of a vector? 2. What are the properties of the cross product of two vectors? 3. How do you find the unit vector of a given vector? 4. What is the geometric interpretation of the dot product? 5. How do you find the area of a parallelogram formed by two vectors? 6. What is the significance of orthogonal vectors in vector spaces? 7. How do you determine if three vectors are coplanar? 8. What is the relationship between the dot product and the angle between two vectors? **Tip:** Remember that the cross product of two vectors results in a vector that is orthogonal to both original vectors.