Let the position vectors of three vertices of a triangle be \( \overrightarrow{p} = 4\hat{i} + \hat{j} - 3\hat{k} \), \( \overrightarrow{q} = -5\hat{i} + 2\hat{j} + 3\hat{k} \), and \( \overrightarrow{r} = -5\hat{i} + 3\hat{j} + 2\hat{k} \). Then \( \alpha + 2\beta + 5\gamma \) is equal to:
Show Hint
For problems involving position vectors and the centroid or orthocenter, use the properties of these points to form relationships between the coefficients and solve for the desired value.
The sum \( \alpha + 2\beta + 5\gamma \) is calculated utilizing the centroid and orthocenter properties with the given triangle vertex position vectors. Final Answer: \( \alpha + 2\beta + 5\gamma = 3 \).
