Sean Fitzpatrick |
---|
University of Lethbridge |
![]() |
Given vectors \(\vec{v}=\la 2,-1,3\ra\) and \(\vec{w} = \la -4,5,1\ra\text{,}\) find:
Recall: Given a real number \(c\) and a vector \(\vec{v}=\la a,b\ra\text{,}\)
Some observations:
For any vector \(\vec{v}\text{,}\) \(0\vec{v}=\vec{0}\) and \((-1)\vec{v}=-\vec{v}\text{.}\)
We have \(\vec{v}+\vec{v}=2\vec{v}\text{.}\)
In general, \(a\vec{v}+b\vec{v}=(a+b)\vec{v}\text{.}\)
Also, \(c(\vec{v}+\vec{w})=c\vec{v}+c\vec{w}\text{.}\)
\(2\vec{v}=\vec{v}+\vec{v}\text{,}\) so \(2\vec{v}\) is in the same direction as \(\vec{v}\text{,}\) but twice as long.
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 \(\R^3\text{:}\)
Write the vector \(\vec{v} = \la 4,-7,6\ra\) in terms of the vectors \(\i, \j, \k\text{.}\)
The dot product provides the algebra – geometry bridge.
Let \(\vec{v}=\la v_1,v_2\ra, \vec{w}=\la w_1,w_2\ra\) be vectors in \(\R^2\text{.}\) The dot product \(\vv\dotp\ww\) is given by
For \(\vec{v}=\la v_1,v_2,v_3\ra, \vec{w}=\la w_1,w_2,w_3\ra\) in \(\R^3\text{,}\) we similarly have
For \(\vec{v}=\la 3,-4\ra, \vec{w}=\la 6,2\ra\text{:}\)
Compute \(\vv\dotp\ww\)
Compute \(\vv\dotp(3\ww)\)
Compute \(3(\vv\dotp\ww)\)
For \(\uu = \la 2,-1,3\ra, \vv=\la -3, -1, 4\ra, \ww = \la 0, 1, -5\ra\text{:}\)
Compute \(\uu\dotp \vv + \uu\dotp\ww\)
Compute \(\uu\dotp (\vv+\ww)\)
Let \(\uu, \vv, \ww\) be vectors, and let \(c\) be a scalar. Then:
\(\vv\dotp\ww = \ww\dotp\vv\)
\(\uu\dotp(\vv+\ww) = \uu\dotp\vv+\uu\dotp \ww\)
\(\uu\dotp (c\vv) = (c\uu)\dotp \vv = c(\uu\dotp\vv)\)
\(\vv\dotp\vv = \norm{\vv}^2\)
\(\vv\dotp\ww = \norm{\vv}\,\norm{\ww}\cos(\theta)\text{,}\) where \(\theta\) is the angle between \(\vv\) and \(\ww\text{.}\)
The dot product lets us compute angles between vectors. Example:
Most useful for us: when \(\theta = \pi/2\text{.}\)
We say that two vectors \(\vv, \ww\) are orthogonal if \(\vv\dotp\ww = 0\text{.}\)
Decide if the triangle with vertices \(P=(1,0,2)\text{,}\) \(Q=(3,-1,0)\text{,}\) \(R = (4, 3, -1)\) is a right-angled triangle.
This is probably the most important application of the dot product in Math 1410. To give it proper attention, we'll hold it over to Thursday's class.
Defined for vectors in \(\R^3\) only.
Produces a vector rather than a scalar.
Cross product \(\vv\times \ww\) is orthogonal to both \(\vv\) and \(\ww\text{.}\)
Definition: if \(\vv = \la v_1,v_2,v_3\ra, \ww = \la w_1,w_2,w_3\ra\text{,}\)
Let \(\uu = \la 3, -1, 2\ra, \vv = \la 0, 2, -1\ra, \ww = \la -2, 0, 4\ra\text{.}\) Find:
\(\uu\times \vv\)
\(\vv\times \ww\)
\(\ww\times \vv\)
If the angle between \(\vv\) and \(\ww\) is \(\theta\text{,}\)
Useful to note: if \(\vv\) and \(\ww\) form 2 of 4 sides of a parallelogram, that parallelogram has area \(A = \norm{\vv}\,\norm{\ww}\sin\theta\text{.}\)
Let \(P=(0,2,-1), Q = (3, 1, -2), R = (4, -2, 0), S = (7, -3, -1)\text{.}\)
Verify that the quadrilateral with these vertices is a parallelogram, and find its area.
What about the triangle \(\Delta PQR\text{?}\)