Topic 3 - Vectors

Week 7 Material

Question 3.1 $\;$ For the vectors \[{\bf a}=4{\bf i}+3{\bf j}+{\bf k},\quad {\bf b}={\bf i}-{\bf j},\quad {\bf c}=5{\bf k},\quad {\bf d}=2{\bf i}+{\bf j}+7{\bf k} \] (where ${\bf i}$, ${\bf j}$, ${\bf k}$ are the standard basis vectors), calculate the following in terms of ${\bf i}$, ${\bf j}$, ${\bf k}$.

$\;\;(a)\;\;{\bf a}+{\bf b}, \hspace{1.4cm}$ $\;\;(b)\;\;{\bf b}+{\bf c},\hspace{1.3cm}$ $\;\;(c)\;\;{\bf d}+{\bf a},\hspace{1.4cm}$ $\;\;(d)\;\;2{\bf a}+3{\bf b},\hspace{1cm}$

$\;\;(e)\;\;2{\bf b}+3{\bf c},\hspace{1cm}$ $\;\;(f)\;\;2{\bf d}-{\bf a},\hspace{0.9cm}$ $\;\;(g)\;\;4{\bf c}-5{\bf b},\hspace{0.9cm}$ $\;\;(h)\;\;{\bf a}-{\bf b}+{\bf c}-2{\bf d}.$

Quick Check

$\;(a)\quad 5{\bf i}+2{\bf j}+{\bf k}.$ $\hspace{1cm}$ $\;(b)\quad {\bf i}-{\bf j}+5{\bf k}.$ $\hspace{1cm}$ $\;(c)\quad 6{\bf i}+4{\bf j}+8{\bf k}.$ $\hspace{1cm}$ $\;(d)\quad 11{\bf i}+3{\bf j}+2{\bf k}.$

$\;(e)\quad 2{\bf i}-2{\bf j}+15{\bf k}.$ $\hspace{1cm}$ $\;(f)\quad -{\bf j}+13{\bf k}.$ $\hspace{1cm}$ $\;(g)\quad -5{\bf i}+5{\bf j}+20{\bf k}.$ $\hspace{1cm}$ $\;(h)\quad -{\bf i}+2{\bf j}-8{\bf k}.$

Question 3.2 $\;$ Given the column vectors ${\bf u}=\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix},\; {\bf v}=\begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix},\; {\bf w}=\begin{pmatrix} -1 \\ 0 \\ 0 \end{pmatrix},\;$ calculate

$\;\;(a)\;\;{\bf u}+{\bf v},\hspace{1.1cm}$ $\;\;(b)\;\;{\bf u}+{\bf w},\hspace{1.45cm}$ $\;\;(c)\;\;2{\bf u}+3{\bf v},\hspace{1.5cm}$ $\;\;(d)\;\;2{\bf u}-3{\bf v},$

$\;\;(e)\;\;{\bf v}-{\bf w},\hspace{1cm}$ $\;\;(f)\;\;2{\bf v}-3{\bf w},\hspace{1cm}$ $\;\;(g)\;\;{\bf u}+{\bf v}+{\bf w},\hspace{1cm}$ $\;\;(h)\;\;{\bf w}-{\bf v}-2{\bf u}.$

Quick Check

$\;(a)\quad \begin{pmatrix} 4 \\ 2 \\ 7 \end{pmatrix},$ $\hspace{1cm}$ $\;(b)\quad \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix},$ $\hspace{1cm}$ $\;(c)\quad \begin{pmatrix} 11 \\ 6 \\ 20 \end{pmatrix},$ $\hspace{1cm}$ $\;(d)\quad \begin{pmatrix} -7 \\ -6 \\ -16 \end{pmatrix},$

$\;(e)\quad \begin{pmatrix} 4 \\ 2 \\ 6 \end{pmatrix},$ $\hspace{1cm}$ $\;(f)\quad \begin{pmatrix} 9 \\ 4 \\ 12 \end{pmatrix}$, $\hspace{1cm}$ $\;(g)\quad \begin{pmatrix} 3 \\ 2 \\ 7 \end{pmatrix},$ $\hspace{1cm}$ $\;(h)\quad \begin{pmatrix} -6 \\ -2 \\ -8 \end{pmatrix}$.

Question 3.3 $\;$ The position vectors of four points $A$, $B$, $C$, $D$ are \[ \overrightarrow{OA}={\bf a},\quad \overrightarrow{OB}={\bf b},\quad \overrightarrow{OC}={\bf a}+2{\bf b},\quad \overrightarrow{OD}=2{\bf a}-3{\bf b}. \] Express the following in terms of ${\bf a}$ and ${\bf b}$:

$\;\;(a)\;\;\overrightarrow{AC},\hspace{1cm}$ $\;\;(b)\;\;\overrightarrow{DB},\hspace{1cm}$ $\;\;(c)\;\;\overrightarrow{BC},\hspace{1cm}$ $\;\;(d)\;\;\overrightarrow{CD}.$

Quick Check

$\;(a)\quad 2{\bf b}$, $\hspace{1cm}$ $\;(b)\quad -2{\bf a}+4{\bf b}$, $\hspace{1cm}$ $\;(c)\quad {\bf a}+{\bf b}$, $\hspace{1cm}$ $\;(d)\quad {\bf a}-5{\bf b}$.

Question 3.4 $\;$ For a triangle $ABC$, let $D$, $E$, $F$ be the midpoints of $BC$, $CA$ and $AB$ respectively.

$\;\;(a)\;\;$ Show that $\overrightarrow{FE}=\frac12\overrightarrow{BC}$.

$\;\;(b)\;\;$ By writing $\overrightarrow{AD}$ as $\overrightarrow{AC}+\frac12\overrightarrow{CB}$, etc., show that $\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}={\bf 0}$.

Quick Check

$(a)\;$ Draw a picture.

$(b)\;$ Show $\overrightarrow{AD}=\overrightarrow{AC}+\overrightarrow{EF}$, find similar expressions for $\overrightarrow{BE}$, $\overrightarrow{CF}$ and add.

Question 3.5 $\;\;$ Prove that the diagonals of a parallelogram bisect one another.

Quick Check

Hint: follow the same method as the example at the end of Section 3.1 in the notes. In other words, write the vectors from the corners of the parallelogram to the intersection as (unknown) scalar multiples of the diagonals and show these multiples equal $1/2$.

Question 3.6 $\;$ The velocity of a boat relative to the water is $3{\bf i}+5{\bf j}$ and that of the water relative to the Earth is ${\bf i}-3{\bf j}$. What is the velocity of the boat relative to the Earth?

Quick Check

The relative velocity is $4{\bf i}+2{\bf j}.$

Question 3.7 $\;$ A meteorologist travelling West at $8\,\textsf{km}\,\textsf{h}^{-1}$ finds that the wind seems to blow directly from the South. On doubling their velocity, they find that it appears to come from the South West. Find the velocity of the wind.

Quick Check

The velocity is $-8{\bf i}+8{\bf j}$.

Question 3.8 $\;$ A particle at the corner of a cube is acted on by forces of $1, 2, 3$ Newtons respectively along the diagonals of the faces of the cube which meet at the particle. Find the magnitude of the resultant force acting on the particle. (The “resultant force” just means the vector sum of the individual forces.)

Quick Check

The force is $5$ Newtons.

Question 3.9 $\;$ For the following subsets of $\mathbb{R}^2$,

$\;\;(a)\;\;S=\left\{\begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 3 \\ 1 \end{pmatrix} \right\},\hspace{1.3cm}$ $\;\;(b)\;\;S=\left\{\begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 3 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 0 \end{pmatrix} \right\},$

$\;\;(c)\;\;S=\left\{\begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} -3 \\ -6 \end{pmatrix} \right\},\hspace{1cm}$ $\;\;(d)\;\;S=\left\{\begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} -3 \\ -6 \end{pmatrix}, \begin{pmatrix} 4 \\ 8 \end{pmatrix}\right\},$

determine if $S$: (i) is linearly independent, (ii) spans $\mathbb{R}^2$, (iii) is a basis of $\mathbb{R}^2$.

Justify your answers.

Quick Check

$(a)\;$ $S$ is linearly independent, it does span $\mathbb{R}^2$, and is a basis.

$(b)\;$ $S$ is not linearly independent, it does span $\mathbb{R}^2$, and is not a basis.

$(c)\;$ $S$ is not linearly independent, it doesn’t span $\mathbb{R}^2$, and is not a basis.

$(d)\;$ $S$ is not linearly independent, it does not span $\mathbb{R}^2$, and is not a basis.

Question 3.10 $\;$ For the following subsets of $\mathbb{R}^3$,

$\;\;(a)\;\;S=\left\{\begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}, \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix} \right\},\hspace{0.2cm}$ $\;\;(b)^*\;\; S=\left\{\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}, \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix} \right\},$

$\;\;(c)\;\;S=\left\{\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix} \right\},\hspace{2.1cm}$ $\;\;(d)^*\;\;S=\left\{\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix}, \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \right\},$

determine if $S$: (i) is linearly independent, (ii) spans $\mathbb{R}^3$, (iii) is a basis of $\mathbb{R}^3$.

Justify your answers.

Quick Check

$\;(a)\;$ $S$ is not linearly independent, it doesn’t span $\mathbb{R}^3$, and is not a basis.

$\;(b)\;$ $S$ is linearly independent, it does span $\mathbb{R}^3$, and is a basis.

$\;(c)\;$ $S$ is linearly independent, it doesn’t span $\mathbb{R}^3$, and is not a basis.

$\;(d)\;$ $S$ is not be linearly independent, it does span $\mathbb{R}^3$, and is not a basis.

Question 3.11 $\!{}^*\;$ For which values of $t\in\mathbb{R}$ is the following set of vectors linearly independent? \[S=\left\{\begin{pmatrix} 2 \\ -3 \\ 4 \end{pmatrix}, \begin{pmatrix} t \\ -2 \\ 2 \end{pmatrix}, \begin{pmatrix} -4 \\ 6 \\ -8 \end{pmatrix} \right\}.\]

Quick Check

$S$ is not linearly independent for any value of $t$.

Question 3.12 $\;$ Suppose that ${\bf u}$, ${\bf v}$ and ${\bf w}$ are vectors in $\mathbb{R}^3$. Show that the set $\{{\bf u}, {\bf v}, {\bf w}\}$ is a basis of $\mathbb{R}^3$ if and only if the set $\{{\bf u}+{\bf v}, {\bf v}+{\bf w}, {\bf w}+{\bf u}\}$ is a basis of $\mathbb{R}^3$.

Quick Check

Hint: show that linear combinations of one set are precisely the same vectors as linear combinations of the other set.

Question 3.13 $\;$ For the vectors ${\bf a}=4{\bf i}+3{\bf j}+{\bf k}$, ${\bf b}={\bf i}-{\bf j}$, ${\bf c}=5{\bf k}$, ${\bf d}=2{\bf i}+{\bf j}+7{\bf k}$, calculate

$\;\;(a)\;\;{\bf a}\cdot{\bf b}, \hspace{0.7cm}$ $\;\;(b)\;\;{\bf a}\cdot({\bf b}+{\bf c}),\hspace{0.7cm}$ $\;\;(c)\;\;({\bf b}-{\bf a})\cdot({\bf b}+{\bf a}),\hspace{0.5cm}$

$\;\;(d)\;\;|{\bf a}|,\hspace{1.05cm}$ $\;\;(e)\;\;|{\bf b}|,\hspace{2cm}$ $\;\;(f)\;\;|{\bf c}|,\hspace{2.2cm}$ $\;\;(g)\;\;|{\bf d}|.$

Quick Check

$\;(a)\quad 1,$ $\hspace{1cm}$ $\;(b)\quad 6,$ $\hspace{1cm}$ $\;(c)\quad -24,$ $\hspace{1cm}$ $\;(d)\quad \sqrt{26},$

$\;(e)\quad \sqrt2,$ $\hspace{1cm}$ $\;(f)\quad 5,$ $\hspace{1cm}$ $\;(g)\quad \sqrt{54}.$

Question 3.14 $\;$ For the vectors in the previous question, calculate the cosines of the angles

$\;\;(a)^*\;\;$ between ${\bf a}$ and ${\bf b},\qquad$ $\;\;(b)\;\;$ between ${\bf b}$ and ${\bf c},\qquad$ $\;\;(c)\;\;$ between ${\bf c}$ and ${\bf d}.$

Quick Check

$\;(a)$ $\dfrac{1}{2\sqrt{13}},$ $\hspace{1cm}$ $\;(b)$ $0,$ $\hspace{1cm}$ $\;(c)$ $\dfrac{7}{3\sqrt6}.$

Question 3.15 $\;$ Find unit vectors along the directions of ${\bf u}=\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, {\bf v}=\begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix}$ and ${\bf w}=\begin{pmatrix} -1 \\ 0 \\ 0 \end{pmatrix}.$

Quick Check

We just need to divide each vector by its length:

$\;(a)\;\;\dfrac{1}{\sqrt2}\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix},\hspace{1cm}$ $\;(b)\;\;\dfrac{1}{7}\begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix},\hspace{1cm}$ $\;(c)\;\;\begin{pmatrix} -1 \\ 0 \\ 0 \end{pmatrix}$.

Question 3.16 $\;\;$ Find the angles and side lengths of the triangle whose vertices have position vectors \[{\bf i}+{\bf j}+{\bf k},\quad 2{\bf i}-{\bf j}-{\bf k},\quad -{\bf i}+{\bf j}-{\bf k}.\]

Quick Check

The lengths are $3$, $\sqrt{13}$, $2\sqrt2.$

The angles are $\arccos\left(\frac{-7}{3\sqrt{13}}\right),$ $\arccos\left(-\frac{3}{\sqrt{26}}\right),$ $\arccos\left(-\frac{1}{3\sqrt2}\right)$.

Week 8 Material

Question 3.17 $\;$ For the vectors ${\bf u}=\begin{pmatrix} 1\\0\\1 \end{pmatrix}$ and ${\bf v}=\begin{pmatrix} 3\\2\\6 \end{pmatrix}$, find

$\;\;(a)\;\;$ the projection of ${\bf u}$ onto ${\bf v}$,

$\;\;(b)^*\;\;$ the projection of ${\bf v}$ onto ${\bf u}$.

Quick Check

$(a)\;$ $\dfrac{9}{49}\begin{pmatrix} 3 \\ 2 \\ 6 \end{pmatrix}.$ $\hspace{1cm}$ $(b)\;$ $\dfrac92\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}.$

Question 3.18 $\;$ Given a non-zero vector ${\bf b}$, show that any vector ${\bf a}$ can be written in a unique way as the sum of a vector ${\bf a}_{\parallel}$ parallel to ${\bf b}$ and a vector ${\bf a}_{\perp}$ perpendicular to ${\bf b}$.

Quick Check

Hint: consider the projection of ${\bf a}$ onto ${\bf b}$ and the non-projected part of ${\bf a}$.

Question 3.19 $\;$ A constant force represented by the vector ${{\bf F}}=3{\bf i}+2{\bf j}+{\bf k}$ acts on a particle as it is displaced through the vector ${\bf d}={\bf i}+2{\bf j}$. Find the work done by the force.

Quick Check

The work done equals $7$ (in appropriate units).

Question 3.20 $\;$ Constant forces of magnitudes 4, 2, and 1 Newtons act on a single particle in the directions of the vectors $6{\bf i}+2{\bf j}+3{\bf k}$, $3{\bf i}-2{\bf j}+6{\bf k}$, and $2{\bf i}-3{\bf j}-6{\bf k}$ respectively. Find the work done as the particle is moved from the point $(2,-1,-3)$ to $(5,-1,1)$, where the units of distance are metres.

Quick Check

The work done is $24\,\textsf{J}.$

Question 3.21 $\;$ If the sum of two forces is equal in magnitude to one of them and perpendicular to it in direction, find the magnitude of the other force and the angle between the two forces.

Quick Check

Show that $|{\bf F}_2|=\sqrt{2}|{\bf F}_1|$ and the angle between the forces is $\theta=\frac{3\pi}{4}.$

Question 3.22 $\;$ Find parametric and Cartesian equations of the straight lines

$\;\;(a)\;\;$ through the point $(1,2,3)$ and parallel to $(5,6,2)$,

$\;\;(b)\;\;$ through the point $(1,0,1)$ and parallel to $(4,3,5)$,

$\;\;(c)\;\;$ through the points $(1,1,1)$ and $(2,5,3)$,

$\;\;(d)\;\;$ through the points $(0,0,0)$ and $(2,1,0)$.

Quick Check

$(a)\;$ Parametric: $\;\;{\bf x} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t\begin{pmatrix} 5 \\ 6 \\ 2 \end{pmatrix}\;\;$ and Cartesian: $\;\;\dfrac{x-1}{5} = \dfrac{y-2}{6} = \dfrac{z-3}{2}.$

$(b)\;$ Parametric: $\;\;{\bf x}=\begin{pmatrix} 1\\0\\1 \end{pmatrix} + t\begin{pmatrix} 4\\3\\5 \end{pmatrix}\;\;$ and Cartesian: $\;\;\dfrac{x-1}{4} = \dfrac{y}{3} = \dfrac{z-1}{5}$.

$(c)\;$ Parametric: $\;\;{\bf x} = \begin{pmatrix} 1\\1\\1 \end{pmatrix} + t\begin{pmatrix} 1\\4\\2 \end{pmatrix}\;\;$ and Cartesian: $\;\;\dfrac{x-1}{1} = \dfrac{y-1}{4} = \dfrac{z-1}{2}.$

$(d)\;$ Parametric: $\;\;{\bf x} = \begin{pmatrix} 0\\0\\0 \end{pmatrix} + t\begin{pmatrix} 2\\1\\0 \end{pmatrix}\;\;$ and Cartesian: $\;\;\dfrac{x}{2} = \dfrac{y}{1}$ and $z=0.$

Question 3.23 $\!{}^*\;$ Find the value of $c$ so that the point $(4, 10, c)$ lies on the straight line that passes through the points $(-2,0,3 )$ and $(1,5,6)$.

Quick Check

Show that $c = 9.$

Question 3.24 $\!{}^*\;$ Find the angle between the normals to the planes $x+y=1$ and $y+z=2$.

Quick Check

The angle is $\pi/3.$

Question 3.25 $\;$ Find a Cartesian equation of a plane passing through $(1,1,1)$, given that a normal vector to the plane makes angles $\pi/3$, $\pi/4$, $\pi/3$ with ${\bf i}$, ${\bf j}$, ${\bf k}$ respectively.

Quick Check

The equation of the plane is $x+\sqrt2y+z=2+\sqrt2.$

Question 3.26 $\;$ Find a Cartesian equation of a plane perpendicular to the line passing through the points $(1,2,3)$ and $(2,4,12)$, given that the plane passes through $(2,3,-7)$.

Quick Check

The equation of the plane is $x+2y+9z=-55.$

Question 3.27 $\!{}^*\;$ Find a Cartesian equation of the plane that passes through $(1,2,3)$ and is parallel to the plane $3x-y+2z=4$. What is the shortest distance between the two planes?

Quick Check

The equation of the plane is $3x-y+2z=7$ and the shortest distance is $\dfrac{3\sqrt{14}}{14}.$

Week 9 Material

Question 3.28 $\;$ For the vectors \[{\bf a}=4{\bf i}+3{\bf j}+{\bf k},\quad {\bf b}={\bf i}-{\bf j},\quad {\bf c}=5{\bf k},\quad {\bf d}=2{\bf i}+{\bf j}+7{\bf k},\] calculate

$\;\;(a)\;\;{\bf a}\times{\bf b},\hspace{1.85cm}$ $\;\;(b)\;\;{\bf a}\times({\bf b}+{\bf c}),\hspace{1cm}$ $\;\;(c)\;\;({\bf b}-{\bf a})\times({\bf b}+{\bf a}),$

$\;\;(d)\;\;{\bf a}\cdot({\bf b}\times{\bf c}),\hspace{1cm}$ $\;\;(e)\;\;{\bf b}\cdot({\bf c}\times{\bf d}),\hspace{1.15cm}$ $\;\;(f)\;\;{\bf b}\cdot({\bf a}\times{\bf c}).$

Quick Check

$(a)\;$ ${\bf i}+{\bf j}-7{\bf k},$ $\hspace{1cm}$ $(b)\;$ $16{\bf i}-19{\bf j}-7{\bf k},$ $\hspace{1cm}$ $(c)\;$ $-2{\bf i}-2{\bf j}+14{\bf k},$

$(d)\;$ $-35,$ $\hspace{1cm}$ $(e)\;$ $-15,$ $\hspace{1cm}$ $(f)\;$ $35.$

Question 3.29 $\;$ A plane contains the three points $(1,1,1)$, $(3,3,2)$ and $(3,-1,-2)$. Find

$\;\;(a)\;\;$ a unit vector normal to the plane,

$\;\;(b)\;\;$ the Cartesian equation of the plane,

$\;\;(c)\;\;$ the shortest distance to the plane from the origin.

Quick Check

$(a)\;$ $\begin{pmatrix} -1/3 \\ 2/3 \\ -2/3 \end{pmatrix}$ (or negative of this), $\hspace{1cm}$ $(b)\;$ $x-2y+2z=1,$ $\hspace{1cm}$ $(c)\;$ $1/3.$

Question 3.30 $\;$ Using the vector product, find the vector equations of the lines

$\;\;(a)\;\;$ through the point $(1,2,3)$ and parallel to $(5,6,2)$,

$\;\;(b)\;\;$ through the point $(1,0,1)$ and parallel to $(4,3,5)$,

$\;\;(c)\;\;$ through the points $(1,1,1)$and $(2,5,3)$,

$\;\;(d)\;\;$ through the points $(0,0,0)$ and $(2,1,2)$,

$\;\;(e)\;\;$ through the point $(2,3,2)$ and perpendicular to the position vectors of $(1,1,0)$ and $(0,1,1)$,

$\;\;(f)\;\;$ through the point $(1,2,3)$ and perpendicular to the position vectors of $(2,1,1)$ and $(-1,-2,3)$.

Quick Check

$(a)\;$ $\begin{pmatrix} 2y-6z+14 \\ -2x+5z-13 \\ 6x-5y+4 \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}.$ $\hspace{2cm}$ $(b)\;$ $\begin{pmatrix} 5y-3z+3 \\ -5x+4z+1 \\3x-4y-3 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}.$

$(c)\;$ $\begin{pmatrix} 2y-4z+2 \\ -2x+z+1 \\ 4x-y-3 \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}.$ $\hspace{2cm}$ $(d)\;$ $\begin{pmatrix} 2y-z \\ -2x+2z \\ x-2y \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}.$

$(e)\;$ $\begin{pmatrix} y+z-5 \\ -x+z \\ -x-y+5 \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}.$ $\hspace{2cm}$ $(f)\;$ $\begin{pmatrix} -3y+7z-15 \\ 3x+5z-18 \\ -7x-5y+17 \end{pmatrix}= \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}.$

Question 3.31 $\;$ Find the shortest distance between the two lines \[{\bf x}=\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}+t\begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix} \qquad\text{and}\qquad {\bf x}=\begin{pmatrix} -1 \\ 3 \\ -3 \end{pmatrix}+t\begin{pmatrix} 4 \\ -1 \\ 2 \end{pmatrix}.\]

Quick Check

The distance is zero - the lines intersect!

Question 3.32 $\;$ Find the shortest distance between the two lines \[{\bf x}=\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}+t\begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix} \qquad\text{and}\qquad {\bf x}=\begin{pmatrix} -1 \\ 2 \\ -3 \end{pmatrix}+t\begin{pmatrix} 2 \\ 6 \\ -4 \end{pmatrix}.\]

Quick Check

The distance is $2\sqrt5.$

Question 3.33 $\;$ By considering the distance between the two straight lines ${\bf x}={\bf a}+t{\bf u}$ and ${\bf x}={\bf b}+t{\bf v}$ where ${\bf u}\times{\bf v}\neq\boldsymbol{0}$, show that the two lines intersect if \[{\bf v}\cdot({\bf a}\times{\bf u})={\bf v}\cdot({\bf b}\times{\bf u}).\]

Quick Check

The lines are not parallel and we have a formula for the distance between them in this case. What does this imply?

Question 3.34 $\;$ Suppose we are given three vectors ${\bf a}$, ${\bf b}$, ${\bf c}$ in $\mathbb{R}^3$. Explain why they are all contained in a plane precisely when ${\bf a}\cdot({\bf b}\times{\bf c})=0$.

Quick Check

Hint: think about the volume of an appropriate parallelepiped.

Question 3.35 $\;$ Differentiate the following expressions with respect to $t$, where ${\bf r}$ is a vector valued function of $t$ and ${\bf a}$, ${\bf b}$ are constant vectors.

$\;\;(a)\;\;{\bf r}\cdot{\bf r},\hspace{1cm}$ $\;\;(b)\;\;({\bf a}\cdot{\bf r}){\bf b},\hspace{1cm}$ $\;\;(c)\;\;|{\bf r}|^2{\bf r},\hspace{2cm}$ $\;\;(d)\;\;{\bf a}\times\dfrac{d{\bf r}}{dt},$

$\;\;(e)\;\;\dfrac{{\bf r}}{|{\bf r}|},\hspace{1.15cm}$ $\;\;(f)\;\;\dfrac{d{\bf r}}{dt}\cdot\dfrac{d{\bf r}}{dt},\hspace{1cm}$ $\;\;(g)\;\;{\bf r}\cdot\left(\dfrac{d{\bf r}}{dt}\times\dfrac{d^2{\bf r}}{dt^2}\right).$

Express your solutions in terms of $\dot{\bf r}$ and/or $\ddot{\bf r}$, as appropriate.

Quick Check

$(a)\;$ $2{\bf r}\cdot\dot{{\bf r}},$ $\hspace{1cm}$ $(b)\;$ $\left({\bf a}\cdot\dot{{\bf r}}\right){\bf b},$ $\hspace{1cm}$ $(c)\;$ $|{\bf r}|^2\dot{{\bf r}}+2({\bf r}\cdot\dot{{\bf r}}){\bf r},$ $\hspace{1cm}$ $(d)\;$ ${\bf a}\times\ddot{{\bf r}},$

$(e)\;$ $\dfrac{1}{|{\bf r}|}\dot{{\bf r}}-\dfrac{1}{|{\bf r}|^3}({\bf r}\cdot\dot{{\bf r}}){\bf r},$ $\hspace{1cm}$ $(f)\;$ $2\dot{{\bf r}}\cdot\ddot{{\bf r}},$ $\hspace{1cm}$ $(g)\;$ ${\bf r}\cdot\left(\dot{{\bf r}}\times\dddot{{\bf r}}\right).$

Question 3.36 $\;$ The position of an ant in the $xy$-plane as a function of time $t$ is \[{\bf r}(t)={\bf i}\cos(\omega t)+{\bf j}\sin(\omega t).\]

$\;\;(a)\;\;$ Show that the ant moves in a circle.

$\;\;(b)\;\;$ Calculate $\dfrac{d{\bf r}}{dt}$, $\dfrac{d^2{\bf r}}{dt^2}$ and $\dfrac{d{\bf r}}{dt}\cdot\dfrac{d^2{\bf r}}{dt^2}$.

$\;\;(c)\;\;$ If the ant has mass $m$, calculate its angular momentum $\displaystyle{\bf L}={\bf r}\times m\frac{d{\bf r}}{dt}$.

Quick Check

$(a)\;$ How are the $x$ and $y$ coordinates related at time $t$?

$(b)\;$ $\dfrac{d{\bf r}}{dt}=-{\bf i}\,\omega\sin(\omega t)+{\bf j}\,\omega\cos(\omega t)$, $\;\dfrac{d^2{\bf r}}{dt^2}=-\omega^2{\bf r}$ and $\;\dfrac{d{\bf r}}{dt}\cdot\dfrac{d^2{\bf r}}{dt^2}=0$.

$(c)\;$ ${\bf L}=m\omega{\bf k}.$

Question 3.37 $\;$ The position of a fly in $\mathbb{R}^3$ as a function of time $t$ is \[{\bf r}(t)=\begin{pmatrix} R\cos(\omega t) \\ R\sin(\omega t) \\ t \end{pmatrix},\] for constants $R$ and $\omega$. Describe the flightpath of the fly and find its velocity and acceleration at time $t$.

Quick Check

The flightpath is a helix (i.e. a corkscrew shape), and \[\dot{{\bf r}}(t)=\begin{pmatrix} -R\omega\sin(\omega t) \\ R\omega\cos(\omega t) \\ 1 \end{pmatrix},\qquad \ddot{{\bf r}}(t)=-R\omega^2\begin{pmatrix} \cos(\omega t) \\ \sin(\omega t) \\ 0 \end{pmatrix}.\]

Question 3.38 $\;$ An electron with position vector ${\bf r}$, charge $e$ and mass $m$ moves in a constant magnetic field ${\bf B}$ according to the equation \[m\frac{d^2{\bf r}}{dt^2}=e\frac{d{\bf r}}{dt}\times {\bf B}.\] Show that the kinetic energy $E=\dfrac{m}{2}\left|\dfrac{d{\bf r}}{dt}\right|^2$ is constant in time. Hint: show that $\dfrac{dE}{dt}=0$.

Quick Check

Hint: differentiate with the product rule and remember when scalar products vanish.

Question 3.39 $\;$ A ray of light travelling in the direction of the vector ${\bf v}$ is reflected in a mirror which has unit normal $\hat{{\bf n}}$. Find the direction vector of the reflected ray. (Draw a picture!)

Quick Check

The reflected ray has direction ${\bf v}'={\bf v}-2({\bf v}\cdot\hat{{\bf n}})\hat{{\bf n}}.$

Week 10 Material

Question 3.40 $\;$ Find the characteristic polynomials and eigenvalues of the following matrices:

$\;\;(a)\;\;\begin{pmatrix} 1 & 0 \\ 7 & 3 \end{pmatrix},\hspace{1.65cm}$ $\;\;(b)\;\;\begin{pmatrix} 1 & 2 \\ 1 & -1 \end{pmatrix},\hspace{2cm}$ $\;\;(c)\;\;\begin{pmatrix} 2 & -4 \\ 4 & -6 \end{pmatrix},$

$\;\;(d)\;\;\begin{pmatrix} 3 & -1 \\ -6 & 2 \end{pmatrix},\hspace{1cm}$ $\;\;(e)\;\;\begin{pmatrix} 1 & -1 & 0 \\ 1 & 3 & -2 \\ 1 & 1 & -1 \end{pmatrix},\hspace{1cm}$ $\;\;(f)\;\;\begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix},$

$\;\;(g)\;\;\begin{pmatrix} 0 & 1 & 2 \\ 1 & 0 & 1 \\ 2 & 1 & 0 \end{pmatrix},\hspace{1cm}$ $\;\;(h)\;\;\begin{pmatrix} 3 & 4 & 2 \\ -1 & 1 & 2 \\ 1 & -2 & -1 \end{pmatrix}.$

Quick Check

$(a)\;$ The eigenvalues are $\lambda=1,3$. $(b)\;$ The eigenvalues are $\lambda=\pm\sqrt{3}$.

$(c)\;$ The eigenvalues are $\lambda=-2,-2$.

$(d)\;$ The eigenvalues are $\lambda=0, 5$.

$(e)\;$ The eigenvalues are $\lambda=0, 1, 2$.

$(f)\;$ The eigenvalues are $\lambda=2, -1, -1$.

$(g)\;$ The eigenvalues are $\lambda=-2, 1+\sqrt{3}, 1-\sqrt{3}$.

$(h)\;$ The eigenvalues are $\lambda=3, +i\sqrt{5}, -i\sqrt{5}$.

Question 3.41 $\;$ Find the eigenvalues and corresponding eigenspaces of the following matrices:

$\;\;(a)\;\;\begin{pmatrix} 3 & 2 \\ 3 & -2 \end{pmatrix},\hspace{1.7cm}$ $\;\;(b)\;\;\begin{pmatrix} 6 & -4 \\ 3 & -1 \end{pmatrix},\hspace{2.3cm}$ $\;\;(c)\;\;\begin{pmatrix} 2 & -3 \\ 3 & -4 \end{pmatrix}$

$\;\;(d)\;\;\begin{pmatrix} -1 & -3 & 3 \\ -3 & -1 & 3 \\ -3 & -3 & 5 \end{pmatrix},\hspace{0.7cm}$ $\;\;(e)\;\;\begin{pmatrix} -2 & -2 & -5 \\ -7 & -3 & -11 \\ 4 & 2 & 7 \end{pmatrix},\hspace{0.7cm}$ $\;\;(f)\;\;\begin{pmatrix} 5 & 6 & -2 \\ -1 & -2 & 3 \\ -1 & -1 & 2 \end{pmatrix},$

$\;\;(g)\;\;\begin{pmatrix} 1 & 1 \\ -2 & -1 \end{pmatrix},\hspace{1.35cm}$ $\;\;(h)\;\;\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \;\;\text{where $\theta$ is not a whole multiple of $\pi$.}$

Quick Check

$(a)\;$ The eigenvalues are $\lambda=-3, 4$ and the corresponding eigenspaces are \[V_{-3}=\operatorname{Span} \left\{\begin{pmatrix} 1 \\ -3 \end{pmatrix}\right\}, \qquad V_{4}=\operatorname{Span} \left\{\begin{pmatrix} 2 \\ 1 \end{pmatrix}\right\}.\]

$(b)\;$ The eigenvalues are $\lambda=2, 3$ and the corresponding eigenspaces are \[V_{2}=\operatorname{Span}\left\{\begin{pmatrix} 1 \\ 1 \end{pmatrix}\right\}, \qquad V_{3}=\operatorname{Span}\left\{\begin{pmatrix} 4 \\ 3 \end{pmatrix}\right\}.\]

$(c)\;$ The eigenvalues are $\lambda=-1, -1$, (i.e. a repeated eigenvalue) and the corresponding eigenspace is \[V_{-1}=\operatorname{Span}\left\{\begin{pmatrix} 1 \\ 1 \end{pmatrix}\right\}.\]

$(d)\;$ The eigenvalues are $\lambda=-1, 2, 2$ and the corresponding eigenspaces are \[V_{-1}=\operatorname{Span}\left\{\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\right\}, \qquad V_{2}=\operatorname{Span}\left\{\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}\right\}.\]

$(e)\;$ The eigenvalues are $\lambda=-1, 1, 2$ and the corresponding eigenspaces are \[V_{-1}=\operatorname{Span}\left\{\begin{pmatrix} -1 \\ -2 \\ 1 \end{pmatrix}\right\},\qquad V_{1}=\operatorname{Span}\left\{\begin{pmatrix} -1 \\ -1 \\ 1 \end{pmatrix}\right\}, \qquad V_{2}=\operatorname{Span}\left\{\begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}\right\}.\]

$(f)\;$ The eigenvalues are $\lambda=-1, 3, 3$ and the corresponding eigenspaces are \[V_{-1}=\operatorname{Span}\left\{\begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix}\right\}, \qquad V_{3}=\operatorname{Span}\left\{\begin{pmatrix} -2 \\ 1 \\ 1 \end{pmatrix}\right\}.\]

$(g)\;$ The eigenvalues are $\lambda=\pm i$ and the corresponding eigenspaces are \[V_{i}=\operatorname{Span}\left\{\begin{pmatrix} -1 \\ 1-i \end{pmatrix}\right\}, \qquad V_{-i}=\operatorname{Span}\left\{\begin{pmatrix} -1 \\ 1+i \end{pmatrix}\right\}.\]

$(h)\;$ The eigenvalues are $\lambda=e^{\pm i\theta}$ and the corresponding eigenspaces are \[V_{e^{i\theta}}=\operatorname{Span}\left\{\begin{pmatrix} i \\ 1 \end{pmatrix}\right\},\qquad V_{e^{-i\theta}}=\operatorname{Span}\left\{\begin{pmatrix} 1 \\ i \end{pmatrix}\right\}.\]

Question 3.42 $\;$ For each of the following matrices $A$, find an invertible matrix $Y$ such that $D=Y^{-1}AY$ is diagonal and compute $A^{10}$.

$\;\;(a)\;\;A=\begin{pmatrix} 2 & -3 \\ 2 & -5 \end{pmatrix},\hspace{1.8cm}$ $\;\;(b)\;\;A=\begin{pmatrix} 5 & 6 \\ -2 & -2 \end{pmatrix},\hspace{1cm}$ $\;\;(c)\;\;A=\begin{pmatrix} 2 & 3 & 6 \\ 6 & 5 & 12 \\ -3 & -3 & -7 \end{pmatrix},$

$\;\;(d)\;\;A=\begin{pmatrix} -1 & -2 & -4 \\ -4 & -3 & -8 \\ 2 & 2 & 5 \end{pmatrix},\hspace{0.5cm}$ $\;\;(e)\;\;A=\begin{pmatrix} -2 & -2 & -5 \\ -7 & -3 & -11 \\ 4 & 2 & 7 \end{pmatrix}.$

Quick Check

$(a)\;$ $Y=\begin{pmatrix} 1 & 3 \\ 2 & 1 \end{pmatrix}$ and $A^{10}= -\frac{1}{5}\begin{pmatrix} 4^{10}-6 & -3\cdot 4^{10}+3 \\ 2\cdot 4^{10}-2 & -6\cdot 4^{10}+1 \end{pmatrix}$.

$(b)\;$ $Y=\begin{pmatrix} 3 & 2 \\ -2 & -1 \end{pmatrix}$ and $A^{10}=\begin{pmatrix}-3+4\cdot 2^{10} & -6+6\cdot 2^{10} \\2-2\cdot 2^{10} & 4-3\cdot 2^{10} \end{pmatrix}.$

$(c)\;$ $Y=\begin{pmatrix} -1 & -1 & -2 \\ -2 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix}$ and $A^{10}=\begin{pmatrix} 2^{10} & 2^{10}-1 & 2\cdot 2^{10}-2 \\ 2\cdot 2^{10}-2 & 2\cdot 2^{10}-1 & 4\cdot 2^{10}-4 \\ -2^{10}+1 & -2^{10}+1 & -2\cdot 2^{10}+3 \end{pmatrix}.$

$(d)\;$ $Y=\begin{pmatrix} 1 & -1 & -2 \\ 2 & 1 & 0 \\ -1 & 0 & 1 \end{pmatrix}$ and $A^{10}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}.$

$(e)\;$ $Y=\begin{pmatrix} 1 & -1 & -1 \\ 3 & -1 & -2 \\ -2 & 1 & 1 \end{pmatrix}$ and $A^{10}=\begin{pmatrix} -2^{10}+2 & 0 &-2^{10}+1\\-3\cdot 2^{10}+3 & 1&-3\cdot 2^{10}+3 \\ 2\cdot 2^{10}-2 & 0 & 2\cdot 2^{10}-1 \end{pmatrix}.$

Question 3.43 $\;$

$\;\;(a)\;\;$ Find a matrix $B$ satisfying $B^3=A$ where $A=\begin{pmatrix} 97 & -70 \\ 105 & -78 \end{pmatrix}$.

$\;\;(b)\;\;$ Find a matrix $B$ satisfying $B^5=A$ where $A=\begin{pmatrix} -67 & 198 \\ -33 & 98 \end{pmatrix}$.

Quick Check

$(a)\;$ $B= \begin{pmatrix} 13 & -10 \\ 15 & -12 \end{pmatrix}$, $\hspace{2cm}$ $(b)\;$ $B= \begin{pmatrix} -7 & 18 \\ -3 & 8 \end{pmatrix}$

Question 3.44 $\;$ Consider the triangular matrix $A=\begin{pmatrix} a & b \\ 0 & d \end{pmatrix}$. Find necessary and sufficient conditions on the entries $a$, $b$, $d$ so that $A$ is not diagonalisable.

Quick Check

$A$ is not diagonalisable precisely when $a=d$ and $b\neq 0$.