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Taking a stab at this one...

What if we use Th. 13.2.5 part d as follows by using vector functions and allowing k to be the radius since that is what the problem tells us.

Does the following show what we want, or am I missing any elements. This is also allowing one of the planes to be zero, is wlog appropriate here?

$$\frac{d}{dt} r(t) \cdot r(t) = \langle k\cos(t), k\sin(t)\rangle \cdot \langle k\cos(t) , k\sin(t)\rangle$$ $$= \langle k\cos(t), k\sin(t)\rangle \cdot \langle -k\sin(t), k\cos(t) \rangle + \langle k\cos(t) , k\sin(t) \rangle \cdot \langle -k\sin(t) , k\cos(t) \rangle$$ $$= -k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t) - k^2 \cos(t) \sin(t) + t^2 \sin(t) \cos(t)$$

Here we can see the answer is zero! What do you think?

Taking a stab at this one...

What if we use Th. 13.2.5 part d as follows by using vector functions and allowing k to be the radius since that is what the problem tells us.

Does the following show what we want, or am I missing any elements. This is also allowing one of the planes to be zero, is wlog appropriate here?

$$\frac{d}{dt} r(t) \cdot r(t) = \langle k\cos(t), k\sin(t)\rangle \cdot \langle k\cos(t) , k\sin(t)\rangle$$ $$= \langle k\cos(t), k\sin(t)\rangle \cdot \langle -k\sin(t), k\cos(t) \rangle + \langle k\cos(t) , k\sin(t) \rangle \cdot \langle -k\sin(t) , k\cos(t) \rangle$$ $$= -k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t) - k^2 \cos(t) \sin(t) + t^2 \sin(t) \cos(t)$$

Here we can see the answer is zero! What do you think?

Taking a stab at this one...

What if we use Th. 13.2.5 part d as follows by using vector functions and allowing k to be the radius since that is what the problem tells us.

Does the following show what we want, or am I missing any elements. This is also allowing one of the planes to be zero, is wlog appropriate here?

$$\frac{d}{dt} r(t) \cdot r(t) = \langle k\cos(t), k\sin(t)\rangle \cdot \langle k\cos(t) , k\sin(t)\rangle$$ $$= \langle k\cos(t), k\sin(t)\rangle \cdot \langle -k\sin(t), k\cos(t) \rangle + \langle k\cos(t) , k\sin(t) \rangle \cdot \langle -k\sin(t) , k\cos(t) \rangle$$ $$= -k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t) - k^2 \cos(t) \sin(t) + t^2 \sin(t) \cos(t)$$

Here we can see the answer is zero! What do you think?

Clarification Added What I wanted to say was that even though the answer is zero, it also shows that the dot product of $\vec{r}$ and $ \vec{r '}$ is zero...$$= \large[-k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)\large] + \large[ - k^2 \cos(t) \sin(t) + t^2 \sin(t) \cos(t)\large]$$

Taking a stab at this one...

What if we use Th. 13.2.5 part d as follows by using vector functions and allowing k to be the radius since that is what the problem tells us.

Does the following show what we want, or am I missing any elements. This is also allowing one of the planes to be zero, is wlog appropriate here?

$$\frac{d}{dt} r(t) \cdot r(t) = \langle k\cos(t), k\sin(t)\rangle \cdot \langle k\cos(t) , k\sin(t)\rangle$$ $$= \langle k\cos(t), k\sin(t)\rangle \cdot \langle -k\sin(t), k\cos(t) \rangle + \langle k\cos(t) , k\sin(t) \rangle \cdot \langle -k\sin(t) , k\cos(t) \rangle$$ $$= -k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t) - k^2 \cos(t) \sin(t) + t^2 k^2 \sin(t) \cos(t)$$

Here we can see the answer is zero! What do you think?

Clarification Added What I wanted to say was that even though the answer is zero, it also shows that the dot product of $\vec{r}$ and $ \vec{r '}$ is zero...$$= \large[-k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)\large] + \large[ - k^2 \cos(t) \sin(t) + t^2 k^2 \sin(t) \cos(t)\large]$$

Taking a stab at this one...

What if we use Th. 13.2.5 part d as follows by using vector functions and allowing k to be the radius since that is what the problem tells us.

Does the following show what we want, or am I missing any elements. This is also allowing one of the planes to be zero, is wlog appropriate here?

$$\frac{d}{dt} r(t) \cdot r(t) = \langle k\cos(t), k\sin(t)\rangle \cdot \langle k\cos(t) , k\sin(t)\rangle$$ $$= \langle k\cos(t), k\sin(t)\rangle \cdot \langle -k\sin(t), k\cos(t) \rangle + \langle k\cos(t) , k\sin(t) \rangle \cdot \langle -k\sin(t) , k\cos(t) \rangle$$ $$= -k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t) - k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)$$

Here we can see the answer is zero! What do you think?

Clarification Added What I wanted to say was that even though the answer is zero, it also shows that the dot product of $\vec{r}$ and $ \vec{r '}$ is zero...$$= \large[-k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)\large] + \large[ \large[\small-k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)\large]\small + \large[\small - k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)\large]$$

Taking a stab at this one...

What if we use Th. 13.2.5 part d as follows by using vector functions and allowing k to be the radius since that is what the problem tells us.

Does the following show what we want, or am I missing any elements. This is also allowing one of the planes to be zero, is wlog appropriate here?

$$\frac{d}{dt} r(t) \cdot r(t) = \langle k\cos(t), k\sin(t)\rangle \cdot \langle k\cos(t) , k\sin(t)\rangle$$ $$= \langle k\cos(t), k\sin(t)\rangle \cdot \langle -k\sin(t), k\cos(t) \rangle + \langle k\cos(t) , k\sin(t) \rangle \cdot \langle -k\sin(t) , k\cos(t) \rangle$$ $$= -k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t) - k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)$$

Here we can see the answer is zero! What do you think?

Clarification Added What I wanted to say was that even though the answer is zero, it also shows that the dot product of $\vec{r}$ and $ \vec{r '}$ is zero...$$= \large[\small-k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)\large]\small + \large[\small - k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)\large]$$

Taking a stab at this one...

What if we use Th. 13.2.5 part d as follows by using vector functions and allowing k to be the radius since that is what the problem tells us.

Does the following show what we want, or am I missing any elements. elements? This is also allowing one of the planes to be zero, is wlog zero. (is WLOG appropriate here?here?)

$$\frac{d}{dt} r(t) \cdot r(t) = \langle k\cos(t), k\sin(t)\rangle \cdot \langle k\cos(t) , k\sin(t)\rangle$$ $$= \langle k\cos(t), k\sin(t)\rangle \cdot \langle -k\sin(t), k\cos(t) \rangle + \langle k\cos(t) , k\sin(t) \rangle \cdot \langle -k\sin(t) , k\cos(t) \rangle$$ $$= -k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t) - k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)$$

Here we can see the answer is zero! What do you think?

Clarification Added What I wanted to say was that even though the answer is zero, it also shows that the dot product of $\vec{r}$ and $ \vec{r '}$ is zero...$$= \large[\small-k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)\large]\small + \large[\small - k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)\large]$$

Taking a stab at this one...

What if we use Th. 13.2.5 part d as follows by using vector functions and allowing k to be the radius since that is what the problem tells us.

Does the following show what we want, or am I missing any elements? This is also allowing one of the planes to be zero. (is WLOG appropriate here?)

$$\frac{d}{dt} r(t) \cdot r(t) = \langle k\cos(t), k\sin(t)\rangle \cdot \langle k\cos(t) , k\sin(t)\rangle$$ $$= \langle k\cos(t), k\sin(t)\rangle \cdot \langle -k\sin(t), k\cos(t) \rangle + \langle k\cos(t) , k\sin(t) \rangle \cdot \langle -k\sin(t) , k\cos(t) \rangle$$ $$= -k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t) - k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)$$

Here we can see the answer is zero! What do you think?

Clarification Added What I wanted to say was that even though the answer is zero, it also shows that the dot product of $\vec{r}$ and $ \vec{r '}$ is zero...$$= \large[\small-k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)\large]\small + \large[\small - k^2 \cos(t) \sin(t) + k^2 \sin(t) \cos(t)\large]$$