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: Recall the parallelogram diagonal bisection property.
In any parallelogram, the diagonals bisect each other, which implies that the sum of the position vectors of opposite vertices are equal: P⃗ + R⃗ = Q⃗ + S⃗ for consecutive vertices P, Q, R, S.

Step 2: Insert the given vertex coordinates.
(î + 2ĵ) + (5î + 7ĵ) = (4î + 6ĵ) + (aî + bĵ). The left side sums to 6î + 9ĵ.

Step 3: Equate the î and ĵ components separately.
For î: a + 4 = 6 → a = 2. For ĵ: b + 6 = 9 → b = 3.

Step 4: Final conclusion.
The unknown coordinates are a = 2 and b = 3.