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Math 1410, Spring 2020

Introduction to Vectors

January 9, 2020

Sean Fitzpatrick
University of Lethbridge

Cartesian coordinates

We will work mainly in two and three dimensions. We write:

  • \(\R^2 = \{(x,y) \,|\, x,y\in \R\}\) — set of all points in the plane

  • \(\R^3 = \{(x,y,z) \,|\, x,y,z\in \R\}\) — set of all points in “space”

  • Can define \(\R^4, \R^5\text{,}\) etc. similarly, but can't draw them.

Distance comes from the Pythagorean theorem. In \(\R^2\text{,}\) the distance from \((x_1,y_1)\) to \((x_2,y_2)\) is

\begin{equation*} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \end{equation*}
In \(\R^3\text{,}\) we just have to add the \(z\)-coordinate:
\begin{equation*} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} \end{equation*}

Vectors

A vector is a “directed line segment”. It has magnitude and direction.

The position of a vector usually doesn't matter.

Given points \(P\) and \(Q\) we write \(\vec{v}=\overrightarrow{PQ}\) for the vector from \(P\) to \(Q\text{.}\) (Tail at \(P\text{,}\) tip at \(Q\text{.}\))

The magnitude of \(\vec{v}\) is just the distance from tip to tail. We write \(\norm{\vec{v}}\) for the magnitude of \(\vec{v}\text{.}\)

Vector notation

We can't do much by just drawing arrows. Let \(P=(x_1,y_1), Q=(x_2,y_2)\) be two points in the plane.

We write \(\vec{v}=\overrightarrow{PQ}\) in component form as \(\vec{v}=\langle x_2-x_1, y_2-y_1\rangle\text{.}\)

Equivalent notation is the column vector \(\vec{v}=\bbm x_2-x_1\\y_2-y_1\ebm\text{.}\)

Magnitude becomes \(\norm{\vec{v}}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\text{.}\)

Examples

Let \(P=(1,2)\) and \(Q=(3,-4)\text{.}\) Find:

  1. \(\overrightarrow{PQ}\)

  2. \(\orr{QP}\)

  3. \(\norm{\orr{PQ}}\)

Exercise: Repeat the above for \(P=(1,-2,4)\) and \(Q=(3,0,-2)\text{.}\)

Exercise: If \(\orr{PQ}=\la 3,-1,5\ra\) and \(Q=(2,2,-3)\text{,}\) what is \(P\text{?}\)

Contexts

Vectors appear frequently in Physics:

  • Displacement

  • Velocity

  • Force

  • etc.

Can think in terms of simple directions: \(\vec{v}=\la 3,4\ra\) tells us to go 3 units right, 4 units up.

Also find vectors as data arrays (but no longer geometric). Meaning of “vector” expands in second linear algebra course.

Adding vectors

To add vectors, we simply add components.

  • In \(\R^2\text{,}\) with \(\vec{v}=\la v_1,v_2\ra, \vec{w}=\la w_1,w_2\ra\text{,}\)

    \begin{equation*} \vec{v}+\vec{w}=\la v_1+w_1,v_2+w_2\ra\text{.} \end{equation*}

  • In \(\R^3\text{,}\) with \(\vec{v}=\la v_1,v_2,v_3\ra, \vec{w}=\la w_1,w_2,w_3\ra\text{,}\)

    \begin{equation*} \vec{v}+\vec{w}=\la v_1+w_1,v_2+w_2,v_3+w_3\ra\text{.} \end{equation*}

Exercise:

  1. Ask me for examples.

  2. Find \(\la 2,4\ra+\la -5,1\ra\text{.}\)

  3. Find \(\la 4,-2,1\ra - \la 3,-7,6\ra\text{.}\) (How do you think we should define subtraction?)

  4. If \(\vec{x}+\la 3,2\ra = \la 5,9\ra\text{,}\) what is \(\vec{x}\text{?}\)

Addition, geometrically

As “arrows”, vector addition follows “tip-to-tail rule”:

  1. Draw \(\vec{v}\text{.}\)

  2. Draw \(\vec{w}\) with its tail at the tip of \(\vec{v}\text{.}\)

  3. Draw an arrow from the tail of \(\vec{v}\) to the tip of \(\vec{w}\text{.}\)

    This is \(\vec{v}+\vec{w}\text{.}\)

Exercise: try this, for \(\vec{v}=\la 2,1\ra\) and \(\vec{w}=\la 3,2\ra\text{.}\)

Exercise: for \(P=(1,0), Q=(2,2), R=(4,5)\text{,}\) compute \(\orr{PQ},\orr{QR},\orr{PR}\text{,}\) and \(\orr{PQ}+\orr{QR}\text{.}\)

Challenge: show why this works in general, for \(P=(x_1,y_1), Q=(x_2,y_2), R=(x_3,y_3)\text{.}\)

Scalar multiplication

  • Exercise: if \(\vec{v}=\la 3,2,4\ra\text{,}\) what is \(\vec{v}+\vec{v}\text{?}\)

    How are the components related to those of \(\vec{v}\text{?}\)

  • We don't have a good way to define multiplication of two vectors. But we can multiply a vector by a number. This is called scalar multiplication.

    Given a real number \(c\) and a vector \(\vec{v}=\la a,b\ra\text{,}\)

    \begin{equation*} c\vec{v} = c\la a,b\ra = \la ca,cb\ra\text{.} \end{equation*}
    The story in \(\R^3\) is similar:
    \begin{equation*} c\la x,y,z\ra = \la cx,cy,cz\ra\text{.} \end{equation*}

Scalar multiplication, geometrically

  • \(2\vec{v}=\vec{v}+\vec{v}\text{,}\) so \(2\vec{v}\) is in the same direction as \(\vec{v}\text{,}\) but twice as long.

  • What does \(-2\vec{v}\) give us?

  • In general, we have:

    Theorem:

    For any vector \(\vec{v}\) and scalar (number) \(c\text{,}\)

    \begin{equation*} \norm{c\vec{v}}=\lvert c\rvert \norm{\vec{v}}\text{.} \end{equation*}

Parallel vectors, unit vectors

  • Definition: Parallel vectors.

    We say that two vectors \(\vec{v}\) and \(\vec{w}\) are parallel if \(\vec{w}=c\vec{v}\) for some scalar \(c\text{.}\)

  • Definition: Unit vector.

    A vector \(\vec{u}\) is a unit vector if \(\norm{\vec{u}}=1\text{.}\)

  • Unit vectors are useful when we care about direction, but not magnitude.

  • Given \(\vec{v}=\la 2,3\ra\text{,}\) what is a unit vector in the direction of \(\vec{v}\text{?}\)

Standard unit vectors

In \(\R^2\text{:}\)

\begin{equation*} \i = \la 1, 0\ra, \quad \j = \la 0,1\ra\text{.} \end{equation*}

In \(\R^3\text{:}\)

\begin{equation*} \i = \la 1,0,0\ra, \quad \j = \la 0,1,0\ra, \quad \k = \la 0,0,1\ra\text{.} \end{equation*}

Properties of vector algebra

Whoa, we have time for this?