Question

If \(A(\overset{⃗}{a})\), \(B(\overset{⃗}{b})\) and \(C(\overset{⃗}{c})\) are vertices of \(△ABC\). Point D divides segment BC internally in the ratio \(2:1\). Point E divides segment AD internally in the ratio \(1:2\), then the position vector of E is ____

• A. \(\frac{3\overset{⃗}{a}+4\overset{⃗}{b}+2\overset{⃗}{c}}{9}\)
• B. \(\frac{6\overset{⃗}{a}+2\overset{⃗}{b}+\overset{⃗}{c}}{9}\)
• C. \(\frac{3\overset{⃗}{a}+2\overset{⃗}{b}+4\overset{⃗}{c}}{9}\)
• D. \(\frac{6\overset{⃗}{a}+\overset{⃗}{b}+2\overset{⃗}{c}}{9}\)

Hint:

Use the section formula twice: first for D on BC, then for E on AD.

Answer 1

The Correct Option is D

Step 1: Find D:
D divides BC internally in the ratio 2:1, so
\[ \vec d=\frac{2\vec c+1\cdot\vec b}{2+1}=\frac{\vec b+2\vec c}3 \]

Step 2: Find E:
E divides AD internally in the ratio 1:2, so E is one third of the way from A to D:
\[ \vec e=\frac{2\vec a+1\cdot\vec d}{1+2}=\frac{2\vec a+\vec d}3 \]

Step 3: Substitute:
\[ \vec e=\frac{2\vec a+\frac{\vec b+2\vec c}3}3=\frac{6\vec a+\vec b+2\vec c}9 \]

Step 4: Check the Options:
The coefficients \(6+1+2=9\), which equals the denominator, as required for a point inside the triangle. This is option (D). Option (B) swaps the weights of B and C, and (A) and (C) do not have 6 as the largest coefficient.

Final Answer:
The position vector of E is \(\dfrac{6\vec a+\vec b+2\vec c}9\), option (D). \[ \boxed{\text{(D)}} \]