Question

$\overline{b}$ and $\overline{c}$ are non-collinear vectors and $\overline{a}$ is a vector such that $(\overline{c}\cdot\overline{c})\overline{a}=\overline{c}$. If $\overline{a}\times(\overline{b}\times\overline{c})+(\overline{a}\cdot\overline{b})\overline{b}=(4-2\beta-sin~\alpha)\overline{b}+(\beta^{2}-1)\overline{c}$, then the values of the scalars $\alpha$ and $\beta$ are

• A. \(\beta=2, \alpha=n\pi+\frac{\pi}{2}, n\in z \)
• B. \(\beta=-1, \alpha=2n\pi+\frac{\pi}{4}, n\in z \)
• C. \(\beta=1, \alpha=(2n+1)\frac{\pi}{2}, n\in z \)
• D. \(\beta=1, \alpha=2n\pi+\frac{\pi}{2}, n\in z \)

Hint:

When a vector equation equates linear combinations of non-collinear vectors, the coefficients must be identical. Also, remember the range of trigonometric functions to constrain variables.

Answer 1

The Correct Option is D

Step 1: Expand the vector triple product.
Using the identity $\vec a\times(\vec b\times\vec c)=(\vec a\cdot\vec c)\vec b-(\vec a\cdot\vec b)\vec c$, the left side becomes \[ (\vec a\cdot\vec c)\vec b-(\vec a\cdot\vec b)\vec c+(\vec a\cdot\vec b)\vec b. \]
Step 2: Group by $\vec b$ and $\vec c$.
This is $\big[(\vec a\cdot\vec c)+(\vec a\cdot\vec b)\big]\vec b-(\vec a\cdot\vec b)\vec c$, set equal to $(4-2\beta-\sin\alpha)\vec b+(\beta^2-1)\vec c$.
Step 3: Use the condition on $\vec a$.
From $(\vec c\cdot\vec c)\vec a=\vec c$ we get $\vec a=\dfrac{\vec c}{|\vec c|^2}$, hence $\vec a\cdot\vec c=1$ and $\vec a\cdot\vec b=\dfrac{\vec b\cdot\vec c}{|\vec c|^2}$, call this $\mu$.
Step 4: Compare coefficients.
Since $\vec b,\vec c$ are non-collinear we can match each: for $\vec c$, $-\mu=\beta^2-1$; for $\vec b$, $1+\mu=4-2\beta-\sin\alpha$.
Step 5: Add the two equations.
The $\mu$ terms cancel, giving $1=(4-2\beta-\sin\alpha)+(\beta^2-1)$, so $\sin\alpha=\beta^2-2\beta+2=(\beta-1)^2+1$.
Step 6: Solve using the range of sine.
Since $\sin\alpha\le1$ but $(\beta-1)^2+1\ge1$, equality forces $(\beta-1)^2=0$, so $\beta=1$ and $\sin\alpha=1$, giving $\alpha=2n\pi+\tfrac{\pi}{2}$. This is option 4.
\[ \boxed{\beta=1,\ \alpha=2n\pi+\tfrac{\pi}{2},\ n\in z} \]