Skip to main content ☰ Contents Index You! < Prev ^ Up Next > \(\newcommand{\avec}{{\mathbf a}}
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Section 5.1 The dot product
Definition 5.1.0.1 .
For two-dimensional vectors \(\vvec\) and \(\wvec\text{,}\) their dot product \(\vvec\cdot\wvec\) is the scalar defined to be
\begin{equation*}
\vvec\cdot\wvec = \twovec{v_1}{v_2}\cdot\twovec{w_1}{w_2} =
v_1w_1 + v_2w_2\text{.}
\end{equation*}
Definition 5.1.0.2 .
For \(n\) -dimensional vectors \(\vvec\) and \(\wvec\text{,}\) their dot product \(\vvec\cdot\wvec\) is the scalar defined to be
\begin{equation*}
\vvec\cdot\wvec =
v_1w_1 + v_2w_2+\cdots + v_nw_n\text{.}
\end{equation*}
Activity 5.1.0.1 .
Suppose that \(\vvec=\twovec{3}{4}\) and \(\wvec=\twovec{2}{-2}\text{.}\)
(a) Compute the dot product \(\vvec\cdot\wvec
\text{.}\)
(b) Sketch the vector \(\vvec\text{.}\) Then use the Pythagorean theorem to find the length of \(\vvec\text{.}\)
(c) Compute the dot product \(\vvec\cdot\vvec\text{.}\) How is the dot product related to the length of \(\vvec\text{?}\)
Definition 5.1.0.3 .
The magnitude , or length, of \(\vvec\text{,}\) which we denote as \(\len{\vvec}\text{,}\) is \(\len{\vvec} = \sqrt{(v_1)^2 + (v_2)^2+\cdots +
(v_n)^2}\text{.}\)
We saw in the first activity that \(\vvec\cdot\vvec = \len{\vvec}^2\text{.}\)
Definition 5.1.0.4 .
Two vectors \(\vvec\) and \(\wvec\) are perpendicular or orthogonal if \(\vvec\cdot\wvec=0\text{.}\)
Activity 5.1.0.2 .
Let \(\vvec=\twovec32\) and \(\wvec=\twovec{-1}3\text{.}\)
(a) Find the lengths \(\len{\vvec}\) and \(\len{\wvec}\) using the dot product.
(b) Find the dot product \(\vvec\cdot\wvec\text{.}\) Are \(\vvec\) and \(\wvec\) orthogonal?
(c) Consider the vector \(\xvec = \twovec{-2}{3}\text{.}\) Find \(\vvec\cdot \xvec\text{.}\) Are \(\vvec\) and \(\xvec\) orthogonal?
(d) For what value of \(k\) is the vector \(\twovec6k\) perpendicular to \(\wvec\text{?}\)
Activity 5.1.0.3 .
Suppose that
\begin{equation*}
\vvec=\fourvec203{-2}, \hspace{24pt}
\wvec=\fourvec1{-3}41\text{.}
\end{equation*}
(a) Find \(\len{\vvec}\text{.}\)
(b) Find \(\len{\wvec}\text{.}\)
(c) Find \(\vvec\cdot\wvec\text{.}\)
