Proving the Bisector Property of Vectors { and }

To prove that the vector bisects the angle between the unit vectors and , we can follow these steps. This proof involves understanding the properties of unit vectors, calculating the angle between the vectors, normalizing the sum of the vectors, and confirming the bisector property using the cosine rule.

Step 1: Understand Unit Vectors

Since and are unit vectors:

Step 2: Calculate the Angle

Let be the angle between vectors and . The cosine of the angle can be expressed using the dot product:

costheta frac{

Step 3: Normalize the Sum of Vectors

The vector is not necessarily a unit vector. To find the direction of this vector, we normalize it:

Step 4: Find the Magnitude of

The magnitude of is calculated as follows:

|

Step 5: Determine the Direction of the Angle Bisector

The angle bisector of the angle between and can be expressed in terms of the unit vectors as:

This is the same expression we have for after normalization.

Step 6: Confirming the Bisector Property

To show that bisects the angle between and , we need to verify that the angles between and and between and are equal. Using the cosine rule:

(costheta_1 frac{mathbf{a} cdot mathbf{c}}{|mathbf{a}| |mathbf{c}|} frac{mathbf{a} cdot left(frac{mathbf{a} - mathbf{b}}{|mathbf{a} - mathbf{b}|}right)}{1} frac{mathbf{a} cdot mathbf{a} - mathbf{a} cdot mathbf{b}}{|mathbf{a} - mathbf{b}|} frac{1 - mathbf{a} cdot mathbf{b}}{|mathbf{a} - mathbf{b}|})

(costheta_2 frac{mathbf{b} cdot mathbf{c}}{|mathbf{b}| |mathbf{c}|} frac{mathbf{b} cdot left(frac{mathbf{a} - mathbf{b}}{|mathbf{a} - mathbf{b}|}right)}{1} frac{mathbf{b} cdot mathbf{a} - mathbf{b} cdot mathbf{b}}{|mathbf{a} - mathbf{b}|} frac{mathbf{a} cdot mathbf{b} - 1}{|mathbf{a} - mathbf{b}|})

Conclusion

Since (costheta_1 costheta_2), it follows that (theta_1 theta_2). Thus, the vector (mathbf{a} - mathbf{b}) indeed bisects the angle between the vectors (mathbf{a}) and (mathbf{b}).

In this proof, we have shown that the vector (mathbf{a} - mathbf{b}) bisects the angle between the unit vectors (mathbf{a}) and (mathbf{b}).