Question

If the points \[ \vec{P}=\vec{i}+2\vec{j},\quad \vec{Q}=4\vec{i}+6\vec{j},\quad \vec{R}=5\vec{i}+7\vec{j},\quad \vec{S}=a\vec{i}+b\vec{j} \] are the consecutive vertices of a parallelogram \(PQRS\), then

• A. \(a=2,\ b=4\)
• B. \(a=3,\ b=4\)
• C. \(a=2,\ b=3\)
• D. \(a=3,\ b=5\)

Hint:

For a parallelogram with consecutive vertices \(P,Q,R,S\), use the diagonal property \(\vec{P}+\vec{R}=\vec{Q}+\vec{S}\).

Answer 1

The Correct Option is C

Step 1: Use the property of diagonals of a parallelogram.
In a parallelogram, diagonals bisect each other.
Therefore, for consecutive vertices \(P,Q,R,S\), \[ \vec{P}+\vec{R}=\vec{Q}+\vec{S}. \]

Step 2: Substitute the given position vectors.
\[ (\vec{i}+2\vec{j})+(5\vec{i}+7\vec{j}) = (4\vec{i}+6\vec{j})+(a\vec{i}+b\vec{j}). \] Simplifying the left side, \[ 6\vec{i}+9\vec{j} = (a+4)\vec{i}+(b+6)\vec{j}. \]

Step 3: Compare coefficients of \(\vec{i}\) and \(\vec{j}\).
Comparing coefficients of \(\vec{i}\), \[ a+4=6. \] \[ a=2. \] Comparing coefficients of \(\vec{j}\), \[ b+6=9. \] \[ b=3. \]

Step 4: Final conclusion.
Therefore, \[ \boxed{a=2,\ b=3} \]