Problem :
Let $a$ and $c$ are unit vectors and $|b| =4$ with $a \times b = 2a \times c$ The angle between $a$ and $c$ is $\arccos (\frac{1}{4})$. If $b-2c =\lambda a$ then find $\lambda$
My approach :
Since $a$ and $c$ are unit vectors $\Rightarrow |a| =|c| = 1$
we have $a.c =|a||c|cos\theta$
$\Rightarrow a.c = 1 . 1 \frac{1}{4}$
Please suggest how to proceed further to get $\lambda$ will be of great help.
Thanks.
Best Answer
Hint
$$b=\lambda a+2c \Rightarrow b \cdot a= \lambda |a|^2 +2a\cdot c \Rightarrow \lambda=a\cdot b-1/2 \quad (1)$$
$$a \times b = 2a \times c \Rightarrow 1\cdot 4\cdot \sin \alpha=2\cdot 1\cdot 1\cdot \sin \theta \quad (2)$$
but $\sin \theta=\sqrt{1-\cos^2\theta}=\sqrt{1-(1/4)^2}=\sqrt{15}/4$ and so $\sin \alpha=\sqrt{15}/8$
$$a\cdot b= \cos \alpha$$