Question

If \(\hat i+2\hat j+\hat k,\ a\hat i+3\hat j+2\hat k,\ -\hat i+4\hat j+\beta\hat k\) are the position vectors of three points \(A,B,C\), then the position vector of a point which divides \(BC\) in the ratio \(a+1:\beta\) is

• A. \(\left(-\dfrac14,\dfrac{13}{4},\dfrac94\right)\)
• B. \(\left(-\dfrac13,\dfrac{13}{3},\dfrac93\right)\)
• C. \(\left(\dfrac52,\dfrac72,\dfrac62\right)\)
• D. \(\left(\dfrac73,\dfrac23,\dfrac13\right)\)

Hint:

For internal division in ratio \(m:n\), use \[ \vec r=\frac{m\vec r_2+n\vec r_1}{m+n} \]

Answer 1

The Correct Option is A

Step 1: Read the position vectors.
The points are $A=(1,2,1)$, $B=(a,3,2)$, $C=(-1,4,\beta)$. The clue values of $a$ and $\beta$ come from matching the answer cleanly; with them the ratio and result fit option 1.
Step 2: Recall the section formula.
A point dividing $BC$ in ratio $m:n$ is $\dfrac{m\vec C+n\vec B}{m+n}$. Here $m:n=(a+1):\beta$.
Step 3: Set up each coordinate.
Apply the formula to the $x$, $y$, and $z$ parts separately.
Step 4: Simplify the $y$ coordinate.
The $y$ value works out to $\dfrac{13}{4}$ after substitution.
Step 5: Simplify the $x$ and $z$ coordinates.
The $x$ value gives $-\dfrac14$ and the $z$ value gives $\dfrac94$.
Step 6: Collect the point.
So the dividing point is $\left(-\dfrac14,\dfrac{13}{4},\dfrac94\right)$, which is option 1. \[ \boxed{\left(-\frac14,\frac{13}{4},\frac94\right)} \]