Subsection1.8.1Scalar or Dot Product of Two Vectors
Subsection1.8.2Vector or Cross Product of Two Vectors
Subsection1.8.3Dot and Cross Product of Cartesian Unit Vectors
Since \(\hat i\text{,}\) \(\hat j\text{,}\) and \(\hat k\text{,}\) has unit magnitude, and are perpendicular to each other, we get the following dot and cross products.
\begin{align}
\text{Dot products: }\amp\hat i \cdot \hat i = 1,\ \ \hat j \cdot \hat j = 1,\ \ \hat k \cdot \hat k = 1,\tag{1.8.1}\\
\amp \hat i \cdot \hat j = 0,\ \ \hat j \cdot \hat k = 0,\ \ \hat k \cdot \hat i = 0,\tag{1.8.2}\\
\text{Cross products: }\amp\hat i \times \hat i = 0,\ \ \hat j \times \hat j = 0,\ \ \hat k \times \hat k = 0,\tag{1.8.3}\\
\amp \hat i \times \hat j = \hat k,\ \ \hat j \times \hat k = \hat i,\ \ \hat k \times \hat k = \hat j,\tag{1.8.4}
\end{align}
With these, you can easily see how the components of an arbitrary vector \(\vec A = \left( A_x, A_y, A_z\right)\) are just the dot product of \(\vec A \) with the unit vectors along the axex.
\begin{equation*}
A_x = \vec A \cdot \hat i,\ \ A_y = \vec A \cdot \hat j,\ \ A_z = \vec A \cdot \hat k.
\end{equation*}
The magnitude of \(\vec A \) is same as before
\begin{equation*}
A = \sqrt{\vec A \cdot \vec A} = \sqrt{A_x^2 + A_y^2 + A_z^2}.
\end{equation*}
Cosines of the angles \(\vec A\) makes with positive axes, called direction cosines of the vector, are