Statement-1: If $\vec a$ and $\vec b$ are two unit vectors, then $|\vec a+\vec b|^2+|\vec a-\vec b|^2=4$
Statement-2: If a and b are two unit vectors inclined at an angle θ, then the unit vectors along their sum and difference are $\frac{\vec a+\vec b}{2 \cos θ/2}$ and $\frac{\vec a-\vec b}{2 \sin θ/2}$ respectively.
Answer & explanation
Correct answer: option 2
We have,
$|\vec a+\vec b|^2+|\vec a-\vec b|^2=2(|\vec a|^2+|\vec b|^2)$
$⇒|\vec a+\vec b|^2+|\vec a-\vec b|^2=2(1+1)=4$
So, statement-1 is true.
Since, $\vec a$ and $\vec b$ are unit vectors inclined at an angle θ.
$∴|\vec a+\vec b|^2+|\vec a|^2+|\vec b|^2+2(\vec a.\vec b)$
$=|\vec b|^2+|\vec b|^2+|\vec a||\vec b|\cos θ$
$⇒|\vec a+\vec b|^2=2+2\cos θ=2(1+\cos θ)=4\cos^2 θ/2$ and, $|\vec a-\vec b|^2=4\sin^2 θ/2$
Therefore, unit vectors along $\vec a+\vec b$ and $\vec a-\vec b$ are $\frac{\vec a+\vec b}{2 \cos θ/2}$ and $\frac{\vec a-\vec b}{2 \sin θ/2}$ respectively.
So, statement-2 is true.