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 ____

Question

If $|\mathbf{a}| = \sqrt{3}, |\mathbf{b}| = 5, |\mathbf{b}||\mathbf{c}| = 10$, the angle between $\mathbf{b}$ and $\mathbf{c}$ is $\pi/3$, and $\mathbf{a}$ is perpendicular to $\mathbf{b} \times \mathbf{c}$, then the value of $|\mathbf{a} \times (\mathbf{b} \times \mathbf{c})|$ is:

Answer 1

Step 1: Weights Method:
A point dividing a segment in ratio $m:n$ has weights $n$ at the first end and $m$ at the second. For D on BC with 2:1: weight 1 at B, 2 at C.

Step 2: Apply Again:
E on AD with ratio 1:2: weight 2 at A, 1 at D. Scale D as $\dfrac{\vec b+2\vec c}{3}$, so E has $\dfrac23\vec a+\dfrac13\cdot\dfrac{\vec b+2\vec c}3$.

Step 3: Simplify:
$=\dfrac{6\vec a}9+\dfrac{\vec b}9+\dfrac{2\vec c}9=\dfrac{6\vec a+\vec b+2\vec c}9$. Option (D).

Final Answer:
Option (D). \[ \boxed{\text{(D)}} \]

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 ____

Show Hint

The Correct Option is D

Solution and Explanation

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