Question

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:

• A. \( 4 \)
• B. \( 6 \)
• C. \( 3 \)
• D. \( 1 \)

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.

Answer 1

The Correct Option is C

Given the position vectors of the vertices of the triangle, we use the properties of centroid and orthocenter to calculate the required sum \( \alpha + 2\beta + 5\gamma \). 
Final Answer: \( \alpha + 2\beta + 5\gamma = 3 \).