Question

If \( \overrightarrow{a}, \overrightarrow{b} \) are two non-collinear vectors, then \( |\overrightarrow{b}| \overrightarrow{a} + |\overrightarrow{a}| \overrightarrow{b} \) represents

• A. a vector parallel to an angle bisector of \( \overrightarrow{a}, \overrightarrow{b} \)
• B. a vector along the difference of the vectors \( \overrightarrow{a}, \overrightarrow{b} \)
• C. a vector along \( \overrightarrow{a} + \overrightarrow{b} \)
• D. a vector outside the triangle having \( \overrightarrow{a}, \overrightarrow{b} \) as adjacent sides

Hint:

When encountering vector expressions of the form \( |\overrightarrow{b}| \overrightarrow{a} + |\overrightarrow{a}| \overrightarrow{b} \), recognize that it represents a vector along the angle bisector of the two given vectors.

Answer 1

The Correct Option is A

We are given the vector expression: \[ \overrightarrow{v} = |\overrightarrow{b}| \overrightarrow{a} + |\overrightarrow{a}| \overrightarrow{b} \] 
Step 1: Understanding the vector sum 
This expression represents a weighted sum of the vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \), where the weights are the magnitudes of the other vector. This type of vector sum is known to produce a vector that is in the direction of the angle bisector of the two vectors. 
Step 2: Geometric Interpretation 
- The vector \( \overrightarrow{v} \) lies in the plane formed by \( \overrightarrow{a} \) and \( \overrightarrow{b} \). - The weights assigned to \( \overrightarrow{a} \) and \( \overrightarrow{b} \) ensure that the resultant vector is directed along the angle bisector of the angle between \( \overrightarrow{a} \) and \( \overrightarrow{b} \). 
Step 3: Conclusion 
Since the given expression aligns with the well-known angle bisector theorem in vector form, the vector \( \overrightarrow{v} \) is parallel to the bisector of the angle between \( \overrightarrow{a} \) and \( \overrightarrow{b} \). Thus, the correct answer is: \[ \boxed{\text{a vector parallel to an angle bisector of } \overrightarrow{a}, \overrightarrow{b}.} \]

Answer 2

Step 1: Understanding the Vector Expression
We are given the vector expression:

\[ \overrightarrow{v} = |\overrightarrow{b}| \, \overrightarrow{a} + |\overrightarrow{a}| \, \overrightarrow{b} \]

This is a linear combination of vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \), where each vector is scaled by the magnitude of the other. Such a combination is symmetric and balances the contributions of both vectors proportionally to their lengths.

Step 2: Geometric Interpretation
- The resultant vector \( \overrightarrow{v} \) lies in the same plane as \( \overrightarrow{a} \) and \( \overrightarrow{b} \).
- Because each vector is multiplied by the magnitude of the other, the resulting vector points along the angle bisector of the two vectors.
- This aligns with the known vector identity for the direction of the angle bisector:

\[ \text{Angle bisector} \propto |\overrightarrow{b}| \, \overrightarrow{a} + |\overrightarrow{a}| \, \overrightarrow{b} \]

Step 3: Conclusion
Based on the geometric and algebraic interpretation, we conclude that the vector \( \overrightarrow{v} \) is parallel to the angle bisector of the angle between vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \).

Final Answer: \[ \boxed{\text{a vector parallel to an angle bisector of } \overrightarrow{a}, \overrightarrow{b}.} \]