Unit Vectors

Step 1: Given conditions

Step 2: Find the magnitude of
The magnitude is: Thus:
Step 3: Find
The dot product is: Since , this simplifies to: Thus: Conclusion:
The value of is .

Step 1: {Recall the dot product formula}
The dot product of two vectors and is: where is the angle between the vectors.
Step 2: {Determine the range of }
Since , the magnitude of varies as: The maximum value of occurs when : The minimum value of occurs when :
Step 3: {Conclude the result}
Thus, .