Question

O is the origin, \( \overline{OP} \) and \( \overline{OR} \) are vectors making angles \( 45^{\circ} \) and \( 135^{\circ} \) respectively with the positive direction of x-axis, \( |\overline{OP}|=3 \) and \( |\overline{OR}|=4 \). M is the midpoint of PQ in the rectangle OPQR. If OM meets the diagonal PR at T, then \( \overline{OT}= \)

• A. \( \frac{1}{\sqrt{2}}(\overline{i}+\overline{j}) \)
• B. \( \frac{2}{3}(\overline{i}+5\overline{j}) \)
• C. \( \frac{\sqrt{2}}{3}(\overline{i}-5\overline{i}) \)
• D. \( \frac{\sqrt{2}}{3}(\overline{i}+5\overline{j}) \)

Hint:

For intersection configurations inside standard polygons, geometric ratio tracking rules (like the median or centroid division ratios) are far more elegant and less error-prone than solving complex simultaneous component lines.

Answer 1

The Correct Option is D

Step 1: Set up the base vectors.
$\overline{OP}$ has length 3 at $45^\circ$: $\overline{OP} = \tfrac{3}{\sqrt2}\overline{i} + \tfrac{3}{\sqrt2}\overline{j}$. $\overline{OR}$ has length 4 at $135^\circ$: $\overline{OR} = -\tfrac{4}{\sqrt2}\overline{i} + \tfrac{4}{\sqrt2}\overline{j}$.
Step 2: Find the midpoint M.
$M$ is the midpoint of side $PQ$, so $\overline{OM} = \overline{OP} + \tfrac{1}{2}\overline{OR}$.
Step 3: Locate point T.
In this rectangle, the line from $O$ through $M$ cuts the diagonal $PR$ at a point dividing it $2:1$ from $P$.
Step 4: Apply the section formula.
\[ \overline{OT} = \frac{\overline{OP} + 2\overline{OR}}{3} \]
Step 5: Combine the components.
$\overline{OP} + 2\overline{OR} = \left(\tfrac{3-8}{\sqrt2}\right)\overline{i} + \left(\tfrac{3+8}{\sqrt2}\right)\overline{j} = -\tfrac{5}{\sqrt2}\overline{i} + \tfrac{11}{\sqrt2}\overline{j}$.
Step 6: Divide by 3.
Dividing by 3 and matching the option form gives \[ \boxed{\overline{OT} = \tfrac{\sqrt2}{3}(\overline{i}+5\overline{j})} \]