Answer: x intercept (4,0) y intercept (0,3)
Step-by-step explanation:
Multiply the second equation by -2 so it would be 4x-14y=6 and the first is equation is also 4x-14y=6
So there is infinitely many solutions
Answer:
![\mathbf{v}(\pi/4)=-3\mathbf{i}+3\mathbf{j}+3 \mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28%5Cpi%2F4%29%3D-3%5Cmathbf%7Bi%7D%2B3%5Cmathbf%7Bj%7D%2B3%20%5Cmathbf%7Bk%7D)
Step-by-step explanation:
<u>Instantaneous Velocity</u>
Given r(t) as the vector function of the position for time t, the instantaneous velocity is computed as:
![\mathbf{v}=\frac{d\mathbf{r}}{dt}=\mathbf{r}'(t)](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%3D%5Cfrac%7Bd%5Cmathbf%7Br%7D%7D%7Bdt%7D%3D%5Cmathbf%7Br%7D%27%28t%29)
We are given:
![\mathbf{r}=\sin^2(3t)\mathbf{i}+3t\mathbf{j}-\cos^2(3t)\mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Br%7D%3D%5Csin%5E2%283t%29%5Cmathbf%7Bi%7D%2B3t%5Cmathbf%7Bj%7D-%5Ccos%5E2%283t%29%5Cmathbf%7Bk%7D)
Thus:
![\mathbf{v}(t)=\mathbf{r}'(t)=[\sin^2(3t)\mathbf{i}+3t\mathbf{j}-\cos^2(3t)\mathbf{k}]'](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28t%29%3D%5Cmathbf%7Br%7D%27%28t%29%3D%5B%5Csin%5E2%283t%29%5Cmathbf%7Bi%7D%2B3t%5Cmathbf%7Bj%7D-%5Ccos%5E2%283t%29%5Cmathbf%7Bk%7D%5D%27)
Computing the derivative:
![\mathbf{v}(t)=2\sin(3t)\cos(3t)(3)\mathbf{i}+3\mathbf{j}+2\cos(3t)\sin(3t)(3)\mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28t%29%3D2%5Csin%283t%29%5Ccos%283t%29%283%29%5Cmathbf%7Bi%7D%2B3%5Cmathbf%7Bj%7D%2B2%5Ccos%283t%29%5Csin%283t%29%283%29%5Cmathbf%7Bk%7D)
![\mathbf{v}(t)=6\sin(3t)\cos(3t)\mathbf{i}+3\mathbf{j}+6\cos(3t)\sin(3t)\mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28t%29%3D6%5Csin%283t%29%5Ccos%283t%29%5Cmathbf%7Bi%7D%2B3%5Cmathbf%7Bj%7D%2B6%5Ccos%283t%29%5Csin%283t%29%5Cmathbf%7Bk%7D)
Evaluating for t=π/4:
![\mathbf{v}(\pi/4)=6\sin(3\pi/4)\cos(3\pi/4)\mathbf{i}+3\mathbf{j}+6\cos(3\pi/4)\sin(3\pi/4) \mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28%5Cpi%2F4%29%3D6%5Csin%283%5Cpi%2F4%29%5Ccos%283%5Cpi%2F4%29%5Cmathbf%7Bi%7D%2B3%5Cmathbf%7Bj%7D%2B6%5Ccos%283%5Cpi%2F4%29%5Csin%283%5Cpi%2F4%29%20%5Cmathbf%7Bk%7D)
![\mathbf{v}(\pi/4)=6(\sqrt{2}/2)(-\sqrt{2}/2)\mathbf{i}+3\mathbf{j}+6(-\sqrt{2}/2))(\sqrt{2}/2)) \mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28%5Cpi%2F4%29%3D6%28%5Csqrt%7B2%7D%2F2%29%28-%5Csqrt%7B2%7D%2F2%29%5Cmathbf%7Bi%7D%2B3%5Cmathbf%7Bj%7D%2B6%28-%5Csqrt%7B2%7D%2F2%29%29%28%5Csqrt%7B2%7D%2F2%29%29%20%5Cmathbf%7Bk%7D)
![\mathbf{v}(\pi/4)=-3\mathbf{i}+3\mathbf{j}-3 \mathbf{k}](https://tex.z-dn.net/?f=%5Cmathbf%7Bv%7D%28%5Cpi%2F4%29%3D-3%5Cmathbf%7Bi%7D%2B3%5Cmathbf%7Bj%7D-3%20%5Cmathbf%7Bk%7D)
3.579 is greater.
It is relatively closer to 4 than the other number. 3 and 3 are equal, 5 and 5 are equal, but 7 is greater than 6. Hope this helps :)