Q.(a) For what values of $\mu$ the system of homogeneous equations $x+y+3z=0$; $4x+3y+\mu z=0$; $2x+y+2z=0$ have:

(a) evaluates the coefficient determinant of a homogeneous $3\times3$ linear system as a function of $\mu$ and reads off the trivial/infinite-solution condition; (b) proves the sine addition formula by the vector (cross-product) method.

(a) Values of $\mu$ for the homogeneous system

1. System: $x+y+3z=0;\ 4x+3y+\mu z=0;\ 2x+y+2z=0$, coefficient matrix $A=\begin{pmatrix}1&1&3\\4&3&\mu\\2&1&2\end{pmatrix}$.
2. A homogeneous system is always consistent (it has the trivial solution $x=y=z=0$). It has only the trivial solution iff $\det A\neq 0$, and it has infinitely many (non-trivial) solutions iff $\det A=0$ (rank $<3$).
3. Expand along the first row: $\det A = 1(3\cdot2-\mu\cdot1)-1(4\cdot2-\mu\cdot2)+3(4\cdot1-3\cdot2)$
4. $=(6-\mu)-(8-2\mu)+3(-2) = 6-\mu-8+2\mu-6 = \mu-8$.
5. (i) Only the trivial solution when $\det A\neq0$, i.e. $\mu\neq 8$.
6. (ii) Infinitely many solutions when $\det A=0$, i.e. $\mu = 8$.

(b) Vector proof of $\sin(A+B)=\sin A\cos B+\cos A\sin B$

1. Let $\hat a=\cos A\,\hat i+\sin A\,\hat j$ be the unit vector making angle $A$ with the positive $x$-axis, and $\hat b=\cos B\,\hat i-\sin B\,\hat j$ be the unit vector making angle $-B$ with the positive $x$-axis (i.e. angle $B$ measured below the axis).
2. The angle swept from $\hat a$ (at $+A$) round to $\hat b$ (at $-B$), going clockwise (the negative sense), has magnitude $A+B$; so as directed angle from $\hat a$ to $\hat b$ it is $-(A+B)$. …