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:
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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.
Using the position vectors of the triangle's vertices and the properties of the centroid and orthocenter, we determine the value of \( \alpha + 2\beta + 5\gamma \). Final Answer: \( \alpha + 2\beta + 5\gamma = 3 \).
