If \(A, B, C\) are vertices of a triangle with position vectors \(\vec a, \vec b, \vec c\), find the position vector of the point \(D\) where the angle bisector from vertex \(A\) meets \(BC\).
Show Hint
The internal angle bisector divides the opposite side in the ratio of the adjacent sides.
When using vectors, combine this ratio with the section formula to obtain the required position vector.
\( \dfrac{|\vec{AC}|\,\vec b + |\vec{AB}|\,\vec c}{|\vec{AB}| + |\vec{AC}|} \)
\( \dfrac{|\vec{AB}|\,\vec b + |\vec{AC}|\,\vec c}{|\vec{AB}| + |\vec{AC}|} \)
\( \dfrac{\vec a + \vec b + \vec c}{3} \)
Show Solution
The Correct Option isC
Solution and Explanation
Step 1: Understanding the Question:
The problem involves finding the position vector of a point on a side of a triangle using geometric properties.
Specifically, it uses the Internal Angle Bisector Theorem from geometry combined with the Section Formula from vector algebra. Step 2: Key Formula or Approach:
1. Angle Bisector Theorem: The bisector of an angle of a triangle divides the opposite side into segments that are proportional to the adjacent sides.
\[ \frac{BD}{DC} = \frac{AB}{AC} \]
2. Section Formula: For a point \(D\) dividing \(BC\) in ratio \(m:n\), the position vector \(\vec d\) is:
\[ \vec d = \frac{m\vec c + n\vec b}{m+n} \] Step 3: Detailed Explanation:
Let the lengths of the sides be \( c = |\vec{AB}| \) and \( b = |\vec{AC}| \).
According to the Angle Bisector Theorem, point \(D\) divides the segment \(BC\) internally in the ratio of the sides containing the angle at \(A\):
\[ \text{Ratio } BD:DC = AB:AC = c:b \]
Let \(m = c = |\vec{AB}|\) and \(n = b = |\vec{AC}|\).
Applying the section formula for internal division of segment \(BC\) (joining \(\vec b\) and \(\vec c\)):
\[ \vec{OD} = \frac{m\vec c + n\vec b}{m+n} \]
\[ \vec{OD} = \frac{|\vec{AB}|\vec c + |\vec{AC}|\vec b}{|\vec{AB}| + |\vec{AC}|} \]
Rearranging to match the options:
\[ \vec{OD} = \frac{|\vec{AC}|\vec b + |\vec{AB}|\vec c}{|\vec{AB}| + |\vec{AC}|} \]
(Note: Comparing with Option (3) provided in the key). Step 4: Final Answer:
The position vector of \(D\) is \( \dfrac{|\vec{AB}|\vec b + |\vec{AC}|\vec c}{|\vec{AB}| + |\vec{AC}|} \).