Sean Fitzpatrick |
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University of Lethbridge |
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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
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 \(\vec{v}\text{.}\)
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 \(\norm{\vec{v}}=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\text{.}\)
Let \(P=(1,2)\) and \(Q=(3,-4)\text{.}\) Find:
\(\overrightarrow{PQ}\)
\(\orr{QP}\)
\(\norm{\orr{PQ}}\)
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 \(P\text{?}\)
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.
To add vectors, we simply add components.
In \(\R^2\text{,}\) with \(\vec{v}=\la v_1,v_2\ra, \vec{w}=\la w_1,w_2\ra\text{,}\)
In \(\R^3\text{,}\) with \(\vec{v}=\la v_1,v_2,v_3\ra, \vec{w}=\la w_1,w_2,w_3\ra\text{,}\)
Exercise:
Ask me for examples.
Find \(\la 2,4\ra+\la -5,1\ra\text{.}\)
Find \(\la 4,-2,1\ra - \la 3,-7,6\ra\text{.}\) (How do you think we should define subtraction?)
If \(\vec{x}+\la 3,2\ra = \la 5,9\ra\text{,}\) what is \(\vec{x}\text{?}\)
As “arrows”, vector addition follows “tip-to-tail rule”:
Draw \(\vec{v}\text{.}\)
Draw \(\vec{w}\) with its tail at the tip of \(\vec{v}\text{.}\)
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 \(P=(x_1,y_1), Q=(x_2,y_2), R=(x_3,y_3)\text{.}\)
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 \(\vec{v}=\la a,b\ra\text{,}\)
\(2\vec{v}=\vec{v}+\vec{v}\text{,}\) 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:
For any vector \(\vec{v}\) and scalar (number) \(c\text{,}\)
We say that two vectors \(\vec{v}\) and \(\vec{w}\) are parallel if \(\vec{w}=c\vec{v}\) for some scalar \(c\text{.}\)
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 \(\vec{v}\text{?}\)
In \(\R^2\text{:}\)
In \(\R^3\text{:}\)
Whoa, we have time for this?