Chapter 11.5, Problem 38E

(a)

To determine

To show: The quantity $B^2-4AC$ is invariant under rotation.

Explanation of Solution

Given information:

A quantity is invariant under rotation if it doesn’t change when the axes are rotated.

Proof:

As the value of $A^{\prime }$ , $B^{\prime }$ and $C^{\prime }$ after rotation is equal to $A{\cos }^2\varphi +B\sin \varphi \cos \varphi +C{\sin }^2\varphi$ , $2\left(C-A\right)\sin \varphi \cos \varphi +B\left({\cos }^2\varphi -{\sin }^2\varphi \right)$ and $A{\sin }^2\varphi -B\sin \varphi \cos \varphi +C{\cos }^2\varphi$ respectively.

Substitute $A{\cos }^2\varphi +B\sin \varphi \cos \varphi +C{\sin }^2\varphi$ , $2\left(C-A\right)\sin \varphi \cos \varphi +B\left({\cos }^2\varphi -{\sin }^2\varphi \right)$ and $A{\sin }^2\varphi -B\sin \varphi \cos \varphi +C{\cos }^2\varphi$ for $A^{\prime }$ , $B^{\prime }$ and $C^{\prime }$ respectively in the expression ${B^{\prime }}^2-4A^{\prime }{C}^{\prime }$ .

${B^{\prime }}^2-4A^{\prime }{C}^{\prime }={\left(2\left(C-A\right)\sin \varphi \cos \varphi +B\left({\cos }^2\varphi -{\sin }^2\varphi \right)\right)}^2-4\times \left(A{\cos }^2\varphi +B\sin \varphi \cos \varphi +C{\sin }^2\varphi \right)\left(A{\sin }^2\varphi -B\sin \varphi \cos \varphi +C{\cos }^2\varphi \right)$$=4{\left(C-A\right)}^2{\sin }^2\varphi {\cos }^2\varphi +B^2{\left({\cos }^2\varphi -{\sin }^2\varphi \right)}^2+2\times 2\left(C-A\right)\sin \varphi \cos \varphi \times B\left({\cos }^2\varphi -{\sin }^2\varphi \right)-4\left(A^2{\cos }^2\varphi {\sin }^2\varphi -AB\sin \varphi {\cos }^3\varphi +AC{\cos }^4\varphi +AB{\sin }^3\varphi \cos \varphi -B^2{\cos }^2\varphi {\sin }^2\varphi +BC\sin \varphi {\cos }^3\varphi +AC{\sin }^4\varphi -BC{\sin }^3\varphi \cos \varphi +C^2{\cos }^2\varphi {\sin }^2\varphi \right)$$=4{\left(C-A\right)}^2{\sin }^2\varphi {\cos }^2\varphi +B^2{\left({\cos }^2\varphi -{\sin }^2\varphi \right)}^2+2\times 2\left(C-A\right)\sin \varphi \cos \varphi \times B\left({\cos }^2\varphi -{\sin }^2\varphi \right)-4\left(A^2{\cos }^2\varphi {\sin }^2\varphi -AB\sin \varphi {\cos }^3\varphi +AC{\cos }^4\varphi +AB{\sin }^3\varphi \cos \varphi -B^2{\cos }^2\varphi {\sin }^2\varphi +BC\sin \varphi {\cos }^3\varphi +AC{\sin }^4\varphi -BC{\sin }^3\varphi \cos \varphi +C^2{\cos }^2\varphi {\sin }^2\varphi \right)$$=A^2\left(4{\sin }^2\varphi {\cos }^2\varphi -4{\sin }^2\varphi {\cos }^2\varphi \right)+B^2\left({\cos }^4\varphi +{\sin }^4\varphi -2{\cos }^2\varphi {\sin }^2\varphi +4{\cos }^2\varphi {\sin }^2\varphi \right)+C^2\left(4{\cos }^2\varphi {\sin }^2\varphi -4{\cos }^2\varphi {\sin }^2\varphi \right)+AB\left(-4\left({\cos }^3\varphi \sin \varphi -{\sin }^3\varphi \cos \varphi \right)+4{\cos }^3\varphi \sin \varphi -4{\sin }^3\varphi \cos \varphi \right)+BC\left(4\left({\cos }^3\varphi \sin \varphi -{\sin }^3\varphi \cos \varphi \right)-4\left({\cos }^3\varphi \sin \varphi -{\sin }^3\varphi \cos \varphi \right)\right)+AC\left(-8{\sin }^2\varphi {\cos }^2\varphi -4{\cos }^4\varphi -4{\sin }^4\varphi \right)$

Further simplify,

  $\begin{array}{c}{B^{\prime }}^2-4A^{\prime }{C}^{\prime }=B^2\left({\cos }^4\varphi +{\sin }^4\varphi +2{\cos }^2\varphi {\sin }^2\varphi \right)-4AC\left(2{\sin }^2\varphi {\cos }^2\varphi +{\cos }^4\varphi +{\sin }^4\varphi \right)\\ =B^2{\left({\cos }^2\varphi +{\sin }^2\varphi \right)}^2-4AC{\left({\cos }^2\varphi +{\sin }^2\varphi \right)}^2\\ =B^2\left(1\right)-4AC\left(1\right)\\ =B^2-4AC\end{array}$

As the value of ${B^{\prime }}^2-4A^{\prime }{C}^{\prime }$ is equal to $B^2-4AC$ .

Hence, the quantity $B^2-4AC$ is invariant under rotation.

(b)

To determine

To show: The quantity $A+C$ is invariant under rotation.

Explanation of Solution

Given information:

A quantity is invariant under rotation if it doesn’t change when the axes are rotated.

Proof:

As the value of $A^{\prime }$ , $B^{\prime }$ and $C^{\prime }$ after rotation is equal to $A{\cos }^2\varphi +B\sin \varphi \cos \varphi +C{\sin }^2\varphi$ , $2\left(C-A\right)\sin \varphi \cos \varphi +B\left({\cos }^2\varphi -{\sin }^2\varphi \right)$ and $A{\sin }^2\varphi -B\sin \varphi \cos \varphi +C{\cos }^2\varphi$ respectively.

Substitute $A{\cos }^2\varphi +B\sin \varphi \cos \varphi +C{\sin }^2\varphi$ and $A{\sin }^2\varphi -B\sin \varphi \cos \varphi +C{\cos }^2\varphi$ for $A^{\prime }$ and $C^{\prime }$ respectively in the expression $A^{\prime }+C^{\prime }$ .

  $\begin{array}{c}{A}^{\prime }+C^{\prime }=A{\cos }^2\varphi +B\sin \varphi \cos \varphi +C{\sin }^2\varphi +A{\sin }^2\varphi -B\sin \varphi \cos \varphi +C{\cos }^2\varphi \\ =A\left({\sin }^2\varphi +{\cos }^2\varphi \right)+B\left(\sin \varphi \cos \varphi -\sin \varphi \cos \varphi \right)+C\left({\sin }^2\varphi +{\cos }^2\varphi \right)\\ =A\left(1\right)+B\left(0\right)+C\left(1\right)\\ =A+C\end{array}$

As the value of $A^{\prime }+C^{\prime }$ is equal to $A+C$ .

Hence, the quantity $A+C$ is invariant under rotation.

(c)

To determine

To check: Whether the quantity $F$ is invariant under rotation or not.

Answer to Problem 38E

Yes, the quantity $F$ is invariant under rotation.

Explanation of Solution

Given information:

A quantity is invariant under rotation if it doesn’t change when the axes are rotated.

The value of the quantity $F^{\prime }$ is always equal to the value of $F$ under any kind of rotation of axis.

Therefore, the quantity $F$ is invariant under rotation.