Definition 3.4. Dot Product.
\begin{equation}
\mathbf u \cdot \mathbf v = u_1v_1+u_2v_2+u_3v_3.\tag{3.5}
\end{equation}
Section 3.2 The Dot Product
The term \(u_1v_1+u_2v_2+u_3v_3\) in the numerator of (3.4) is known as the dot product of the two vectors \(\mathbf u = \langle u_1,u_2,u_3\rangle\) and \(\mathbf v = \langle v_1,v_2,v_3\rangle\text{,}\) and is denoted \(\mathbf u \cdot \mathbf v\text{.}\)
Remark 3.5. Orthogonal vectors.
If the two vectors \(\mathbf u\) and \(\mathbf v\) are orthogonal, then \(\theta = \pi/2\text{,}\) which means \(\mathbf u \cdot \mathbf v = 0\text{.}\) Conversely, if \(\mathbf u\) and \(\mathbf v\) are two vectors such that \(\mathbf u \cdot \mathbf v = 0\text{,}\) then \(\mathbf u\) and \(\mathbf v\) are orthogonal.
Fact 3.6. Orthogonality Criterion.
\begin{equation}
\mathbf u \cdot \mathbf v = 0 \iff \mathbf u \perp \mathbf v.\tag{3.6}
\end{equation}
Example 3.7. A right triangle detected with the dot product.
Consider the triangle with vertices \(A(1,-1)\text{,}\) \(B(1,1)\text{,}\) and \(C(2,0)\text{.}\) Show that this is a right triangle using the dot product.
Solution.
\begin{equation*}
\begin{aligned}
\vec{CA} \amp = \langle 1-2,\,-1-0\rangle = \langle -1,-1\rangle \\
\vec{CB} \amp = \langle 1-2,\,1-0\rangle = \langle -1,1\rangle \\
\vec{CA}\cdot\vec{CB} \amp = 1 - 1 = 0 \implies \vec{CA} \perp \vec{CB}.
\end{aligned}
\end{equation*}
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Example 3.9. The remaining angles of the triangle.
For the triangle of Example 3.7 (shown again in Figure 3.8), calculate the remaining angles.
Solution.
For the angle between the sides \(\vec{AC}\) and \(\vec{AB}\text{,}\) we have
\begin{equation*}
\begin{aligned}
\vec{AC} \amp = \langle 2-1,\,0+1\rangle = \langle 1,1\rangle \implies |\vec{AC}| = \sqrt2 \\
\vec{AB} \amp = \langle 1-1,\,1+1\rangle = \langle 0,2\rangle \implies |\vec{AB}| = 2 \\
\vec{AB}\cdot\vec{AC} \amp = 0 + 2 = 2 \\
\cos\theta \amp = \frac{\vec{AB}\cdot\vec{AC}}{|\vec{AB}||\vec{AC}|}
= \frac{2}{2\sqrt2} = \frac{1}{\sqrt2} \implies \theta = \frac{\pi}{4}.
\end{aligned}
\end{equation*}
Subsection 3.2.1 Properties of the Dot Product
Fact 3.10.
If \(\mathbf u\text{,}\) \(\mathbf v\text{,}\) and \(\mathbf w\) are any vectors and \(c\) is a scalar, then
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\(\displaystyle \mathbf u \cdot \mathbf v = \mathbf v \cdot \mathbf u\)
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\(\displaystyle (c\mathbf u)\cdot \mathbf v = \mathbf u \cdot (c\mathbf v) = c(\mathbf u \cdot \mathbf v)\)
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\(\displaystyle \mathbf u \cdot (\mathbf v + \mathbf w) = \mathbf u \cdot \mathbf v + \mathbf u \cdot \mathbf w\)
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\(\displaystyle \mathbf u \cdot \mathbf u = |\mathbf u|^2\)
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\(\displaystyle \mathbf 0 \cdot \mathbf u = 0\)
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