Question

For two unit vectors which of which of the following relation is true ?

• A. $(a\times b{)}^2-(a.b{)}^2=1$
• B. $(a\times b{)}^2+(a.b{)}^2=1$
• C. $(a\times b{)}^2+(a.b{)}^2=2$
• D. $(a\times b{)}^2+(a.b{)}^2=3$

Answer 1

The correct option is B $(a\times b{)}^2+(a.b{)}^2=1$

Let $a$ and $b$ be unit vectors and $\theta$ be angle between them

$(a\times b{)}^2=|a{|}^2|b{|}^2{\sin }^2\theta ={\sin }^2\theta$

$(a.b{)}^2=|a{|}^2|b{|}^2{\cos }^2\theta =co{s}^2\theta$

$\Rightarrow (a\times b{)}^2+(a.b{)}^2={\sin }^2\theta +{\cos }^2\theta =1$

Therefore option $B$ is correct