Question

PQRS is a quadrilateral and \( \vec{PQ} = \vec{a} \), \( \vec{QR} = \vec{b} \), \( \vec{SP} = \vec{a} - \vec{b} \).
\( M \) is the midpoint of \( QR \), and \( X \) lies on \( \vec{SM} \) such that \( \vec{SX} = \frac{4}{5} \vec{SM} \).
If \( \vec{SM} = m(4\vec{a} - \vec{b}) \), and \( \vec{SX} = n(4\vec{a} - \vec{b}) \), then \( m + n = \):

• A. \( \frac{9}{10} \)
• B. \( \frac{10}{9} \)
• C. \( \frac{11}{9} \)
• D. \( \frac{4}{3} \)

Hint:

Vector Ratios}
When two vectors share direction, scalar multiples help define proportions
Use midpoint and point-ratio logic effectively
Match coefficients to solve for unknowns

Answer 1

The Correct Option is A

Given \( \vec{SX} = \frac{4}{5} \vec{SM} \Rightarrow n = \frac{4}{5}m \) So: \[ m + n = m + \frac{4}{5}m = \frac{9}{5}m \] From given: \[ \vec{SM} = m(4\vec{a} - \vec{b}), \quad \vec{SX} = \frac{4}{5} \vec{SM} = \frac{4}{5} m(4\vec{a} - \vec{b}) = n(4\vec{a} - \vec{b}) \] Thus, \[ n = \frac{4}{5}m \Rightarrow m + n = \frac{9}{10} \]