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Q.
If $a \times(b \times c)+(a \cdot b) b=(4-2 \beta-\sin \alpha) b+\left(\beta^{2}-1\right) c$ and $(c \cdot c) a=c$, while $b$ and $c$ are non-collinear, then
Vector Algebra
Solution:
We have, $a \times(b \times c)+(a \cdot b) b$
$=(4-2 \beta-\sin \alpha) b+\left(b^{2}-1\right) c\,\,\, (1)$
and $(c \cdot c) a=c\,\,\, (2)$
where $b$ and $c$ are non-collinear vectors and $\alpha, \beta$ are scalars
From (2), $ (c \cdot c) a \cdot c=c \cdot c $
$ \therefore a \cdot c=1 \,\,\,\, (3)$
From (1), we get
$(a \cdot c) b-(a \cdot b) c+(a \cdot b) b$
$=(4-2 \beta-\sin \alpha) b+\left(\beta^{2}-1\right) c$
or $[1+(a \cdot b)] b-(a \cdot b) c$
$=(4-2 \beta-\sin \alpha) b+\left(\beta^{2}-1\right) c$
$\Rightarrow 1+(a \cdot b)=4-2 \beta-\sin \alpha\,\,\,\, (4)$
and $a \cdot b=-\left(b^{2}-1\right)\,\,\,\, (5)$
$\therefore \sin \alpha=1+(1-\beta)^{2} $
$\Rightarrow \beta=1, \sin \alpha=1$
i.e., $\alpha=\frac{\pi}{2}+2 n \pi, n \in I$