Question

If the sum of two unit vectors is a unit vector, then the magnitude of their difference is...

• A. \(\sqrt{2}\)
• B. \(0\)
• C. \(\sqrt{5}\)
• D. \(\sqrt{3}\)

Hint:

Use |a+b|^2 to find a.b and then compute |a-b|^2.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Let \(\vec a,\vec b\) be unit vectors with \(|\vec a+\vec b|=1\).

Step 2: Find the dot product
\(|\vec a+\vec b|^2=1+1+2\vec a\cdot\vec b=1\), so \(\vec a\cdot\vec b=-\dfrac12\).

Step 3: Find the difference
\(|\vec a-\vec b|^2=1+1-2\vec a\cdot\vec b=2+1=3\).

Step 4: Take the root
\(|\vec a-\vec b|=\sqrt3\).

Final Answer:
The magnitude of the difference is \(\sqrt3\). \[ \boxed{\text{(D) }\sqrt{3}} \]