There are three vectors x,y and z. And between both vectors x,y and y,z are 60 . Given that vectors x,y and z are coplanars. A vector k makes an angle m and n with vectors x and y respectively . We have to find angle between vectors z and k?
I tried it alot ,but could not solve it.
$\newcommand{\vec}[1]{\mathbf{#1}}$Without loss of generality assume that all vectors are unit. We have $$ \vec x \cdot \vec y = \frac{1}{2}, \qquad \vec y \cdot \vec z = \frac{1}{2} $$ Since $\vec x,\vec y,\vec z$ are co-planar, $\vec y = \vec x + \vec z$. (Draw a diagram to see this) We also have $$ \vec k \cdot \vec x = \cos m, \qquad \vec k \cdot \vec y = \cos n $$ Hence $$ \vec k \cdot \vec z = \vec k \cdot (\vec y - \vec x) = \cos n - \cos m $$ Taking $\cos^{-1}$ on both sides finishes the proof.