Length and Angle Measures

Find the length of each vector.

• (a)  $\begin{pmatrix}3\\ 1\end{pmatrix}$     (b)  $\begin{pmatrix}-1\\ 2\end{pmatrix}$     (c)  $\begin{pmatrix}4\\ 1\\ 1\end{pmatrix}$     (d)  $\begin{pmatrix}0\\ 0\\ 0\end{pmatrix}$     (e)  $\begin{pmatrix}1\\ -1\\ 1\\ 0\end{pmatrix}$

Find the angle between each two, if it is defined.

• (a)  $\begin{pmatrix}1\\ 2\end{pmatrix},\begin{pmatrix}1\\ 4\end{pmatrix}$     (b)  $\begin{pmatrix}1\\ 2\\ 0\end{pmatrix},\begin{pmatrix}0\\ 4\\ 1\end{pmatrix}$     (c)  $\begin{pmatrix}1\\ 2\end{pmatrix},\begin{pmatrix}1\\ 4\\ -1\end{pmatrix}$

During maneuvers preceding the Battle of Jutland, the British battle cruiser Lion moved as follows (in nautical miles): $1.2$ miles north, $6.1$ miles $38$ degrees east of south, $4.0$ miles at $89$ degrees east of north, and $6.5$ miles at $31$ degrees east of north. Find the distance between starting and ending positions. (Ignore the earth’s curvature.)

Find $k$ so that these two vectors are perpendicular.

Describe the set of vectors in $\mathbb{R}^{3}$ orthogonal to the one with entries $1$, $3$, and $-1$.

• Find the angle between the diagonal of the unit square in $\mathbb{R}^{2}$ and any one of the axes.

• Find the angle between the diagonal of the unit cube in $\mathbb{R}^{3}$ and one of the axes.

• Find the angle between the diagonal of the unit cube in $\mathbb{R}^{n}$ and one of the axes.

• What is the limit, as $n$ goes to $\infty$, of the angle between the diagonal of the unit cube in $\mathbb{R}^{n}$ and any one of the axes?

Is any vector perpendicular to itself?

Describe the algebraic properties of dot product.

• Is it right-distributive over addition: $(\vec{u}+\vec{v})\mathbin{\smash{\bullet}}\vec{w}=\vec{u}\mathbin{\smash{\bullet}}\vec{w}+\vec{v}\mathbin{\smash{\bullet}}\vec{w}$?

• Is it left-distributive (over addition)?

• Does it commute?

• Associate?

• How does it interact with scalar multiplication?

As always, you must back any assertion with a suitable argument.

Verify the equality condition in Corollary 2.6, the Cauchy-Schwarz Inequality.

• Show that if $\vec{u}$ is a negative scalar multiple of $\vec{v}$ then $\vec{u}\mathbin{\smash{\bullet}}\vec{v}$ and $\vec{v}\mathbin{\smash{\bullet}}\vec{u}$ are less than or equal to zero.

• Show that $\lvert\vec{u}\mathbin{\smash{\bullet}}\vec{v}\rvert=\lvert\vec{u}\,\rvert\,\lvert\vec{v}\,\rvert$ if and only if one vector is a scalar multiple of the other.

Suppose that $\vec{u}\mathbin{\smash{\bullet}}\vec{v}=\vec{u}\mathbin{\smash{\bullet}}\vec{w}$ and $\vec{u}\neq\vec{0}$. Must $\vec{v}=\vec{w}$?

Does any vector have length zero except a zero vector? (If “yes”, produce an example. If “no”, prove it.)

Find the midpoint of the line segment connecting $(x_{1},y_{1})$ with $(x_{2},y_{2})$ in $\mathbb{R}^{2}$. Generalize to $\mathbb{R}^{n}$.

Show that if $\vec{v}\neq\vec{0}$ then $\vec{v}/\lvert\vec{v}\,\rvert$ has length one. What if $\vec{v}=\vec{0}$?

Show that if $r\geq 0$ then $r\vec{v}$ is $r$ times as long as $\vec{v}$. What if $r<0$?

A vector $\vec{v}\in\mathbb{R}^{n}$ of length one is a unit vector. Show that the dot product of two unit vectors has absolute value less than or equal to one. Can ‘less than’ happen? Can ‘equal to’?

Show that the line segments $\overline{(a_{1},a_{2})(b_{1},b_{2})}$ and $\overline{(c_{1},c_{2})(d_{1},d_{2})}$ have the same lengths and slopes if $b_{1}-a_{1}=d_{1}-c_{1}$ and $b_{2}-a_{2}=d_{2}-c_{2}$. Is that only if?

Is $\lvert\vec{u}_{1}+\cdots+\vec{u}_{n}\rvert\leq\lvert\vec{u}_{1}\rvert+\cdots+\lvert\vec{u}_{n}\rvert$? If it is true then it would generalize the Triangle Inequality.

What is the ratio between the sides in the Cauchy-Schwarz inequality?

Why is the zero vector defined to be perpendicular to every vector?

Describe the angle between two vectors in $\mathbb{R}^{1}$.

Give a simple necessary and sufficient condition to determine whether the angle between two vectors is acute, right, or obtuse.

Generalize to $\mathbb{R}^{n}$ the converse of the Pythagorean Theorem, that if $\vec{u}$ and $\vec{v}$ are perpendicular then $\lvert\vec{u}+\vec{v}\,\rvert^{2}=\lvert\vec{u}\,\rvert^{2}+\lvert\vec{v}\,\rvert^{2}$.

Show that $\lvert\vec{u}\,\rvert=\lvert\vec{v}\,\rvert$ if and only if $\vec{u}+\vec{v}$ and $\vec{u}-\vec{v}$ are perpendicular. Give an example in $\mathbb{R}^{2}$.

Show that if a vector is perpendicular to each of two others then it is perpendicular to each vector in the plane they generate. (Remark. They could generate a degenerate plane — a line or a point — but the statement remains true.)

Verify that the definition of angle is dimensionally correct: (1) if $k>0$ then the cosine of the angle between $k\vec{u}$ and $\vec{v}$ equals the cosine of the angle between $\vec{u}$ and $\vec{v}$, and (2) if $k<0$ then the cosine of the angle between $k\vec{u}$ and $\vec{v}$ is the negative of the cosine of the angle between $\vec{u}$ and $\vec{v}$.

Show that the inner product operation is linear: for $\vec{u},\vec{v},\vec{w}\in\mathbb{R}^{n}$ and $k,m\in\mathbb{R}$, $\vec{u}\mathbin{\smash{\bullet}}(k\vec{v}+m\vec{w})=k(\vec{u}\mathbin{\smash{\bullet}}\vec{v})+m(\vec{u}\mathbin{\smash{\bullet}}\vec{w})$.

A ship is sailing with speed and direction $\vec{v}_{1}$; the wind blows apparently (judging by the vane on the mast) in the direction of a vector $\vec{a}$; on changing the direction and speed of the ship from $\vec{v}_{1}$ to $\vec{v}_{2}$ the apparent wind is in the direction of a vector $\vec{b}$.

Find the vector velocity of the wind.

Verify the Cauchy-Schwarz inequality by first proving Lagrange’s identity:

and then noting that the final term is positive. This result is an improvement over Cauchy-Schwarz because it gives a formula for the difference between the two sides. Interpret that difference in $\mathbb{R}^{2}$.