Teaching

Problem 1.

Consider the following planes in $\mathbb{R}^{3}$, where $C$ is constant:

$2Cx - 3y + (C + 4)z = 5$

$(C + 1)x + Cy - z = 1$

A) Find the value(s) of $C$ that makes these planes orthogonal.

B) Explain why if the planes are orthogonal at one intersection point they are orthogonal at all intersection points.

Problem 2.

Consider the following vectors in $\mathbb{R}^{3}$:

$\mathbf{u} = \begin{pmatrix} 1 \ 0 \

1 \end{pmatrix}\ \ \ \ \ :\ \ \ \mathbf{v} = \begin{pmatrix} 2 \
1 \ 3 \end{pmatrix}\ \ \ \ :\ \ \ \mathbf{w} = \begin{pmatrix}
2 \ 0 \ 2 \end{pmatrix}\ \ \ \ \ \ \ \ $

A) Compute the cross product $\mathbf{u} \times \mathbf{v}$.

B) Use a scalar-triple-product to compute the volume of the “parallelopiped” generated by these three vectors $\mathbf{u,\ v,\ w}$.

Problem 3.

Consider the following hyperplanes in $\mathbb{R}^{4}$:

${x_{1} - 2x_{2} + 5x_{3} + x_{4} = 8 }{3x_{1} + 6x_{2} + x_{3} + 4x_{4} = 17}$

Identify vectors orthogonal to each hyperplane and use these to show that the hyperplanes are orthogonal.

Problem 4.

Consider the following planes in $\mathbb{R}^{3}$:

${2x - 3y + z = 5 }{3x + y - 2z = - 1}$

Identify vectors orthogonal to each plane and use these to compute a vector that is tangent to both planes.

Problem 5.

Consider the following vector in $\mathbb{R}^{5}$:

$\mathbf{v} = \begin{pmatrix} 2 \

1 \ 3 \
1 \ 1 \end{pmatrix}\ $

A) Give an example of a nonzero vector that is orthogonal to $\mathbf{v}$.

B) Compute the angle between the vector $\mathbf{v}$ and the basis vector ${\widehat{e}}_{1}$.

Problem 6.

Consider the following vectors in $\mathbb{R}^{4}$:

$\mathbf{u} = \begin{pmatrix} 2 \ 1 \

2 \ 0 \end{pmatrix}\ \ \ :\ \mathbf{v} = \begin{pmatrix} 2 \ 1 \ 2 \
4 \end{pmatrix}\ $

A) Compute the angle between these two vectors..

B) Find a vector that is orthogonal to both $\mathbf{u}$ and $\mathbf{v}$.

Problem 7.

Consider the following vectors in $\mathbb{R}^{4}$:

$\mathbf{u} = \begin{pmatrix} 4 \

5 \ 2 \
2 \end{pmatrix}\ \ \ :\ \mathbf{v} = \begin{pmatrix} 3 \ 0 \ 4 \ 0 \end{pmatrix}\ $

A) Compute the angle between these two vectors.

B) Compute the projected length of $\mathbf{u}$ onto the “$\mathbf{v}$-axis”.

Problem 8.

Consider the following vectors in $\mathbb{R}^{3}$:

$\mathbf{u} = \begin{pmatrix} 3 \

1 \ 2 \end{pmatrix}\ \ \ \ :\ \ \ \ \mathbf{v} = \begin{pmatrix}
1 \ 1 \ 4 \end{pmatrix}\ \ \ \ :\ \ \ \ \mathbf{w} = \begin{pmatrix} 2 \ 0 \
3 \end{pmatrix}$

A) Without computing the angles between them, figure out which pair of vectors, above, has the largest and smallest angle between them.

B) Compute the volume of the parallelopiped spanned by $\mathbf{u}$ and $\mathbf{v}$ and $\mathbf{w}$. Show work.

Problem 9.

Consider the following vectors in $\mathbb{R}^{4}$

$\mathbf{a} = \begin{pmatrix} 0 \

1 \ 0 \ 2 \end{pmatrix}\ \ \ :\ \ \mathbf{b} = \begin{pmatrix} 0 \ 3 \
2 \
1 \end{pmatrix}\ \ \ :\ \ \mathbf{c} = \begin{pmatrix} 1 \
3 \ 2 \ 0 \end{pmatrix}\ \ \ :\ \mathbf{d} = \ \begin{pmatrix}
1 \ 1 \ 0 \ 2 \end{pmatrix}\ \ \ :\ \ \mathbf{e} = \begin{pmatrix} 1 \ 2 \ 0 \ 4 \end{pmatrix}\ \ \ $

A) Which of the above vectors have a positive dot product with $\mathbf{c}$?

B) Is there a pair of vectors from $\mathbf{a},\ \mathbf{b},\ \mathbf{c},\ \mathbf{d},\ \mathbf{e}$ that is orthogonal?

C) Give an example of a nonzero vector that is orthogonal to both $\mathbf{a}$ and $\mathbf{b}$.

Problem 10.

Consider the following four vectors in $\mathbb{R}^{3}$:

$\mathbf{a} = \begin{pmatrix} 1 \ 0 \

3 \end{pmatrix}\ \ :\ \ \mathbf{b} = \begin{pmatrix} 0 \ 2 \ 5 \end{pmatrix}\ \ :\ \ \mathbf{c} = \begin{pmatrix} 1 \ 3 \ 0 \end{pmatrix}\ \ :\ \ \mathbf{d} = \begin{pmatrix} 4 \ 1 \ 0 \end{pmatrix}\ $

A) Is there a pair of orthogonal vectors among the above? Explain.

B) Which three of the vectors above span a parallelopiped with the largest volume?

Problem 11.

Consider the following two curves in $\mathbb{R}^{3}$:

$\gamma_{1}(s) = \begin{pmatrix} s^{2} - 3s \ e^{s} - 1 \ 1 - \cos{2s} \end{pmatrix}\ \ \ \ :\ \ \ \ \gamma_{2}(t) = \begin{pmatrix} sin(t - 1) \ t^{2} + t - 2 \ 1 - \sqrt{t} \end{pmatrix}$

A) Verify that these curves intersect at the origin for some values of $s$ and $t$.

B) At what angle do these curves intersect at the origin?

C) Find a vector that is orthogonal to both curves at the origin.

Problem 12.

Consider the parametrized curve in 3-D given by

$\gamma(t) = \begin{pmatrix} t^{2} - t + 4 \ t^{3} - 3t^{2} + 2t - 1 \ 2t \end{pmatrix}\begin{matrix} \leftarrow x \ \leftarrow y \ \leftarrow z \end{matrix}$

A) Compute the velocity vector of this curve.

B) Write down the equation of a plane orthogonal to this curve at $\gamma(0)$.

C) At what angle does this curve cross the $(x,y)$ plane $z = 0$ ? Explain your reasoning and give your answer as best you can without a calculator…

Problem 13.

Consider the parameterized surface in 3-D given by

$S\begin{pmatrix} u \ v \end{pmatrix} = \left( \begin{array}{r} 4 \ 0 \ 3 \end{array} \right) + \left( \begin{array}{r} u^{2} \

3u \ 2u \end{array} \right) + \left( \begin{array}{r} 0 \
v^{2} \ 5v \end{array} \right)$

Note that the point $P = (5, - 4,10)$ is on this surface (at $u = v = 1$).

A) Find two nonparallel vectors tangent to the surface at this point $P$.

B) Find a nonzero vector perpendicular (i.e., orthogonal) to the surface at this point $P$.

Problem 14.

Consider the parametrized curve in $\mathbb{R}^{4}$ given by

$\gamma(s) = \begin{pmatrix} 3 - s \ s^{2} + 2s \ (s + 1)^{- 1} \

s \end{pmatrix}$

A) Compute the velocity vector to this curve at the point where $s = - 2$.

B) Compute the acceleration vector to this curve at the point where $s = - 2$.

C) Are the velocity and acceleration vectors at this point (where $s = - 2$) orthogonal? Why or why not?

Problem 15.

Consider the following curve in $\mathbb{R}^{3}$:

$\gamma(t) = \begin{pmatrix} (2t - 1)^{2} \ t^{3} - 3t + 4 \ 3t - 2 \end{pmatrix}\ \ \ \ \ :\ \ \ \ 0 \leq t \leq 2$

A) Compute the velocity vector of this curve.

B) Compute the unit tangent vector at the point where $t = 1$.

C) Set up but do not solve an integral to compute the arclength of this curve.

Problem 16.

Consider the following curve in $\mathbb{R}^{6}$:

$\gamma(t) = \begin{pmatrix} 5\cos t \ 4t \ 2t - 1 \ 5 - 2t \ 3\sin t \

4\sin t \end{pmatrix}\ \ \ \ \ :\ \ \ \ 0 \leq t \leq \pi$

A) Compute the velocity vector of this curve.

B) The length of the velocity vector is constant: what is it?

C) Compute the arclength of this curve in $\mathbb{R}^{6}$.

Problem 17.

Consider the following curve in $\mathbb{R}^{4}$, where $C$ is constant:

$\gamma(t) = \begin{pmatrix} 5\cos t \ C\sin t \ 4\cos t \

6\sin t \end{pmatrix}\ \ \ \ \ :\ \ \ \ 0 \leq t \leq \pi$

A) Compute the velocity and acceleration vectors of this curve.

B) Compute and simply (if you can) the arclength element $d\mathcal{l}$ for this curve.

C) The arclength element is constant for some value(s) of $C$: for what value(s)?

Problem 18.

Consider the parametrized curve in 3-D given by

$\gamma(t) = \begin{pmatrix} 3 - t \ t^{3} + t - 2 \ t^{2} - 3t + 4 \end{pmatrix}\begin{matrix} \leftarrow x \ \leftarrow y \ \leftarrow z \end{matrix}$

A) Compute the velocity vector of this curve and evaluate at $t = 1$.

B) Compute the acceleration vector of this curve and evaluate at $t = 1$.

C) At the point $\gamma(1) = (2,0,2)$, the velocity and acceleration vectors lie in a plane. Compute a nonzero vector that is orthogonal to this plane.

Problem 19.

Consider the curve given by

$\gamma(t) = \begin{pmatrix} t^{2} - 5t + 4 \ t(t - 1)(t - 2) \ e^{t - 1} - 1 \end{pmatrix}$

A) Compute the velocity and acceleration vectors to this curve and evaluate these at the point where $\gamma$ intersects the origin.

B) Compute an implicit equation for a plane in $\mathbb{R}^{3}$ orthogonal to the curve $\gamma$ at the origin in the form _______ $x\ +$ _______ $y\ +$ _______ $z\ =$ _______.

Problem 20.

Consider the parametrized path

$\gamma(t) = \begin{pmatrix} \ln{(1 + t)} \ e^{3t} \ \sin{(2t)} \end{pmatrix}$

A) Compute the velocity ($\mathbf{v}$) and acceleration ($\mathbf{a}$) vectors to this curve.

B) Compute the cross product $\mathbf{v}(0) \times \mathbf{a}(0)$.

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