If the points with position vectors $20\hat i + p\hat j,5\hat i -\hat j$ and $10\hat i-13\hat j$ are collinear, then p =

-37

Let $20\hat i + p\hat j,5\hat i -\hat j$ and $10\hat i-13\hat j$ be the position vectors of points P, Q and R respectively. Then,

$\vec{PQ}=-15\hat i-(p + 1)\hat j, \vec{QR}=5\hat i-12\hat j$

It is given that the points P, Q, R are collinear.

$∴\vec{PQ}=λ \vec{QR}$ for some $λ$

$⇒-15\hat i-(p+1)\hat j=2(5\hat i-12\hat j)$

$⇒-15=5λ$ and $-(p + 1) = -12λ$

$⇒λ=-3$ and $p+1=12λ = p = -37$