Exam 13: Vector Functions

Reparametrize the curve with respect to arc length measured from the point where $t = 0$ in the direction of increasing $t$ . $\mathbf { r } ( t ) = ( 7 + 3 t ) \mathbf { i } + ( 10 + 9 t ) \mathbf { j } - ( 6 t ) \mathbf { k }$

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = e ^ { t } \{ \cos 6 t , \sin 6 t , 0 \}$

Find the limit. $\lim _ { t \rightarrow 0 ^ { + } } \langle 8 \cos t , 24 \sin t , 5 t \ln t \rangle$

Find the velocity, acceleration, and speed of an object with position function $\mathbf { r } ( t ) = 4 \sin t \mathbf { i } + \cos t \mathbf { j }$ for $t = \frac { \pi } { 4 }$ . Sketch the path of the object and its velocity and acceleration vectors.

A projectile is fired with an initial speed of $700 \mathrm {~m} / \mathrm { s }$ and angle of elevation $60 ^ { \circ }$ . Find the range of the projectile.

The helix $\mathbf { r } _ { 1 } ( t ) = 4 \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }$ intersects the curve $\mathbf { r } _ { 2 } ( t ) = ( 4 + t ) \mathbf { i } + 6 t ^ { 2 } \mathbf { j } + 5 t ^ { 3 } \mathbf { k }$ at the point $( 4,0,0 )$ . Find the angle of intersection.

Sketch the curve of the vector function $\mathbf { r } ( t ) = 3 t \mathbf { i } + ( 3 t + 8 ) \mathbf { j } , - 1 \leq t \leq 2$ , and indicate the orientation of the curve.

Find the speed of a particle with the given position function. $\mathbf { r } ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } - e ^ { - 4 t } \mathbf { k }$

The torsion of a curve defined by $\mathbf { r } ( t )$ is given by $\tau = \frac { \left( \mathbf { r } ^ { t } \times \mathbf { r } ^ { tt } \right) \cdot \mathbf { r } ^ {t} } { \left| \mathbf { r } ^ { t } \times \mathbf { r } ^ { t } \right| ^ { 2 } }$ Find the torsion of the curve defined by $\mathbf { r } ( t ) = \cos 2 t \mathbf { i } + \sin 2 t \mathbf { j } + 5 t \mathbf { k }$ .

Find the length of the curve $\mathbf { r } ( t ) = - 2 t \mathbf { i } - t \mathbf { j } + t \mathbf { k } , - 2 \leq t \leq 1$ .

Reparametrize the curve with respect to arc length measured from the point where $t = 0$ in the direction of increasing $t$ . $\mathbf { r } ( t ) = ( 5 + 3 t ) \mathbf { i } + ( 8 + 9 t ) \mathbf { j } - ( 6 t ) \mathbf { k }$

Sketch the curve of the vector function $r ( t ) = 5 \sin t \mathbf { i } + 6 \cos t \mathbf { j }$ , and indicate the orientation of the curve.

A particle moves with position function $\mathbf { r } ( t ) = \left( 21 t - 7 t ^ { 3 } - 5 \right) \mathbf { i } + 21 t ^ { 2 } \mathbf { j }$ . Find the tangential component of the acceleration vector.

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 5 t \mathbf { i } + \left( t ^ { 2 } + 5 \right) \mathbf { j }$ .

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 3 \sin t \mathbf { i } + 3 \cos t \mathbf { j } + 5 t \mathbf { k }$ .

A mortar shell is fired with a muzzle speed of 250 ft/sec.Find the angle of elevation of the mortar if the shell strikes a target located 1100 ft away.Round your answer to 2 decimal places.

$\text { A particle moves with position function } \mathbf { r } ( t ) = 5 \cos t \mathbf { i } + 5 \sin t \mathbf { j } + 5 t \mathbf { k } \text {. }$ Find the normal component of the acceleration vector.

Let $C$ be a smooth curve defined by $\mathbf { r } ( t ) = 2 \mathbf { i } + 3 t \mathbf { j } + 2 t ^ { 2 } \mathbf { k }$ , and let $\mathrm { T } ( t )$ and $\mathrm { N } ( t )$ be the unit tangent vector and unit normal vector to $C$ corresponding to $t$ . The plane determined by $\mathbf { T }$ and $\mathbf { N }$ is called the osculating plane. Find an equation of the osculating plane of the curve described by $\mathbf { r } ( t )$ at $t = 1$ .

Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid $x ^ { 2 } + 5 y ^ { 2 } + 5 z ^ { 2 } = 25$ and the parabolic cylinder $y = x ^ { 2 }$ .