Definition 1.5.1.
Let \(\vec{v},\vec{w}\in\mathbb{R}^n\text{,}\) then the dot product of \(\vec{v}\) with \(\vec{w}\) is the scalar value given by
\begin{equation*}
\vec{v}\cdot\vec{w}= v_1 w_1+v_2 w_2+\ldots+v_n w_n
\end{equation*}
Section 1.5 Vectors and Vector Calculations
In order to distinguish vectors (as being more than just \(n\) by \(1\) matrices), we will use the arrow notation and lower case symbols like \(\vec{u}\) and \(\vec{v}\) to denote vectors. Unless otherwise stated, we will use column vectors. For instance, if \(\vec{v} =\begin{bmatrix}v_1\\ v_2\\\vdots\\ v_m\end{bmatrix}\text{,}\) then the second component of \(\vec{v}\) is the scalar \(v_2\text{.}\) The size of a vector in \(\mathbb{R}^n\) is the number of components the vector has. In later work, we will deal with a much more general notion of vectors that will not have components like vectors in \(\mathbb{R}^n\text{.}\) Recall that two vectors in \(\mathbb{R}^n\) are equal if and only if all of their components are equal.
Geometrically, we will view vectors in \(\mathbb{R}^n\) as an arrow which change in the \(i\)-th coordinate being given by the \(i\)-th component of the vector. For instance, the vector \(\langle 1,2,3\rangle\) will have a plot in 3-dimensions that looks like
Vectors do not have a particular beginning or ending point so all of the blue vectors in the following plot are representations of \(\langle 1,2,3\rangle\text{.}\)
Vectors can be added together to measure the net change (done by completing one vector, then the other). Algebraically, vector addition is done componentwise. If \(\vec{v}=\langle v_1,v_2,v_3\rangle\) and \(\vec{w}=\langle w_1,w_2,w_3\rangle\text{,}\) then \(\vec{v}+\vec{w}=\langle v_1+w_1,v_2+w_2,v_3+w_3\rangle\text{.}\)
Many other vector operations can be done componentwise, such as scalar multiplication and subtraction. If \(k\in \mathbb{R}\) and \(\vec{v}\in \mathbb{R}^n\text{,}\) then \(k \vec{v} = \langle k v_1, k v_2, \ldots , k v_n \rangle\text{.}\) If \(\vec{v},\vec{w}\in\mathbb{R}^n\text{,}\) then \(\vec{v}-\vec{w}=\langle v_1-w_1,v_2-w_2,\ldots,v_n-w_n\rangle\text{.}\) Geometrically, scalar multiplication will stretch (and flip if \(k \lt 0\)) the arrow for a vector.
Subsection 1.5.1 The Dot Product
Sometimes we are interested in the total amount of change and not what direction a vector is in. The magnitude (or length) of a vector \(\vec{v}\in\mathbb{R}^n\) is given by the following
\begin{equation*}
\Vert{\vec{v}}\Vert=\sqrt{\sum_{j=1}^{n} (v_i)^2}= \sqrt{\vec{v}\cdot\vec{v}}
\end{equation*}
A unit vector is a vector of length 1.
Definition 1.5.2.
Let \(\vec{v},\vec{w}\in\mathbb{R}^n\text{,}\) then the vectors \(\vec{v}\) and \(\vec{w}\) are orthogonal if \(\vec{v} \cdot \vec{w} =0\text{.}\)
Activity 1.5.1.
Use the Desmos interact embeded below to change the end points of our two vectors. Based on what you see about the value of the dot product for different configurations of vectors, answer the questions below
(a)
How does changing the length of the vectors change the value of the dot product?
(b)
How does changing the angle of the vectors change the value of the dot product?
Definition 1.5.3.
The projection of a vector \(\vec{v}\) onto a vector \(\vec{w}\) gives the vector part of \(\vec{v}\) that is parallel to \(\vec{w}\) and is computed as
\begin{equation*}
proj_{\vec{w}} \vec{v} = \frac{\vec{v}\cdot\vec{w}}{\vec{w}\cdot\vec{w}} \vec{w} = \frac{\vec{v}\cdot\vec{w}}{\Vert\vec{w}\Vert} \left(\frac{\vec{w}}{\Vert\vec{w}\Vert}\right)
\end{equation*}
This is sometimes called the vector projection. The right most part of the definition of the projection vector shows that the projection of \(\vec{v}\) onto \(\vec{w}\) will be a scalar (\(\frac{\vec{v}\cdot\vec{w}}{\Vert\vec{w}\Vert} \))times the unit vector in the direction of \(\vec{w}\text{.}\) The scalar \(\frac{\vec{v}\cdot\vec{w}}{\Vert\vec{w}\Vert} \) is referred to as the scalar projection of \(\vec{v}\) onto \(\vec{w}\text{.}\)
The vector projection of \(\vec{v}\) onto \(\vec{w}\) measures the vector part of \(\vec{v}\) that is parallel to \(\vec{w}\text{,}\) where as the scalar projection of \(\vec{v}\) onto \(\vec{w}\) measures the length of \(\vec{v}\) that is parallel to \(\vec{w}\text{.}\) If you subtract the projection of \(\vec{v}\) onto \(\vec{w}\) from \(\vec{v}\) (\(\vec{v} -proj_{\vec{w}} \vec{v}\)), the result will the part of \(\vec{v}\) that is orthogonal to \(\vec{w}\) because \(\vec{v} -proj_{\vec{w}} \vec{v}\) has subtracted out ALL of \(\vec{v}\) that is parallel to \(\vec{w}\text{.}\)
Theorem 1.5.4.
If \(\theta\) is the angle between two vectors \(\vec{v}\) and \(\vec{w}\text{,}\) then
\begin{equation*}
\vec{v}\cdot\vec{w} = \Vert\vec{v}\Vert \Vert\vec{w}\Vert \cos(\theta)
\end{equation*}
Question 1.5.5.
Let \(\vec{u}=\langle 3,1,2 \rangle\text{,}\) \(\vec{v}=\langle -1,2,0 \rangle\text{,}\) and \(\vec{w}=\langle 4,-3,-1 \rangle\text{.}\)
- \(\displaystyle ||\vec{w}||=\)
- Find a unit vector that is in the opposite direction of \(\vec{w}\)
- \(\displaystyle 3\vec{u}+\vec{v}-2\vec{w}=\)
- Can you choose \(a\) and \(b\) such that \(a\vec{u}+ b\vec{v}=\vec{w}\text{?}\)
- Does \(\vec{w} \cdot \vec{v} \cdot \vec{u}\) make sense? Why or why not?
- What angle does \(\vec{v}\) make with \(\vec{u}\text{?}\)
- What angle does \(\vec{v}\) make with the \(z\)-axis?
- How much of \(\vec{v}\) is parallel to \(\vec{u}\text{?}\)
- How much of \(\vec{u}\) is parallel to \(\vec{v}\text{?}\)
- How much of \(\vec{w}\) is parallel to \(\vec{u}\text{?}\)
- How much of \(\vec{u}\) is orthogonal to \(\vec{v}\text{?}\)
- Find a unit vector that is orthogonal to \(\vec{v}= \langle-1,2,5\rangle\text{.}\) How many such vectors are there?
