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Question 02 a) Two ships, named $S_1$ and $S_2$, are travelling. They both appeared on the light screen of a tracking station at the same time. The distances are measured in hundreds of meters and the time $t$, in seconds, is measured from the instant they were both observed on the light screen of the tracking station. The coordinates of $S_1$ and $S_2$, relative to a fixed origin $O$, are given by $s_1: r_1 = (2t - 4)i + (t - 15)j + (t + 5)k$ $s_2: r_2 = 10i + (-2t + 6)j + (2t - 2)k$ I. Show that $S_1$ and $S_2$ are travelling in perpendicular directions to each other. Given that $S_1$ and $S_2$ continue to travel according to the above vector equations. II. Show further that $S_1$ and $S_2$ will eventually collide at some point $P$, and further determine the coordinates of $P$. III. Calculate, to the nearest meter, the distance between $S_1$ and $S_2$, when they were first observed by the tracking station. b) Let $r(t)$ be a vector function. I. Find the unit tangent vector for the given vector function. $r(t) = < \ln(6t), e^{1-t}, 5t >$ II. Find the tangent line to the vector function at the given point. $r(t) = < 2t, \cos^2(t), e^{6t} >$ at $t=0$ III. Find the unit normal and the binormal vectors for the given vector function. $r(t) = < \cos(2t), \sin(2t), 3 >$ IV. Determine $\nabla f$ for the given function $f$ in the indicated direction. I. $f(x,y) = \ln(2xy) - \sin(x^2 + y^2)$ in the direction of $\vec{v} = (7,3)$ II. $f(x,y) = \cos(x/y)$ in the direction of $\vec{v} = (3,-4)$ III. Find the maximum rate of change of $f(x,y,z) = e^{2x} \cos(y - 2x)$ at a point $(4,-2,0)$ and the direction in which this maximum rate of change occur.

Name: question 02 a two ships named s1 and s2 are travelling they both appeared on the light screen of a tracking station at the same time the distances are measured in hundreds of meters and th 24914
Uploaded: 2024-05-14T11:52:50-08:00
Duration: 2 min 48 s
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Description: question 02 a two ships named s1 and s2 are travelling they both appeared on the light screen of a tracking station at the same time the distances are measured in hundreds of meters and th 24914

          Question 02

a) Two ships, named $S_1$ and $S_2$, are travelling. They both appeared on the light screen of a tracking station at the same time. The distances are measured in hundreds of meters and the time $t$, in seconds, is measured from the instant they were both observed on the light screen of the tracking station. The coordinates of $S_1$ and $S_2$, relative to a fixed origin $O$, are given by
$s_1: r_1 = (2t - 4)i + (t - 15)j + (t + 5)k$
$s_2: r_2 = 10i + (-2t + 6)j + (2t - 2)k$

I. Show that $S_1$ and $S_2$ are travelling in perpendicular directions to each other.
Given that $S_1$ and $S_2$ continue to travel according to the above vector equations.
II. Show further that $S_1$ and $S_2$ will eventually collide at some point $P$, and further determine the coordinates of $P$.
III. Calculate, to the nearest meter, the distance between $S_1$ and $S_2$, when they were first observed by the tracking station.

b) Let $r(t)$ be a vector function.
I. Find the unit tangent vector for the given vector function.
$r(t) = < \ln(6t), e^{1-t}, 5t >$
II. Find the tangent line to the vector function at the given point.
$r(t) = < 2t, \cos^2(t), e^{6t} >$ at $t=0$
III. Find the unit normal and the binormal vectors for the given vector function.
$r(t) = < \cos(2t), \sin(2t), 3 >$

IV. Determine $\nabla f$ for the given function $f$ in the indicated direction.
I. $f(x,y) = \ln(2xy) - \sin(x^2 + y^2)$ in the direction of $\vec{v} = (7,3)$
II. $f(x,y) = \cos(x/y)$ in the direction of $\vec{v} = (3,-4)$
III. Find the maximum rate of change of $f(x,y,z) = e^{2x} \cos(y - 2x)$ at a point $(4,-2,0)$ and the direction in which this maximum rate of change occur.

question 02 a two ships named s1 and s2 are travelling they both appeared on the light screen of a tracking station at the same time the distances are measured in hundreds of meters and th 24914

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