Vector Formulas and Rules
\( \)\( \)\( \)Dot Product
\( \vec a \) and \( \vec b \) are vectors given by their components as follows:
\( \vec a = \langle a_1, a_2 , a_3 \rangle \) and \( \vec b = \langle b_1, b_2 , b_3 \rangle \)
The dot product of vectors \( \vec a \) and \( \vec b \) is defined by:
\( \vec a \cdot \vec b = || \vec a || \; || \vec b || \cos \theta = \sum_{i=1}^{3} a_i b_i = a_1 b_1 + a_2 b_2 + a_3 b_3 \)
where \( \theta \) is the angle between vectors \( \vec a \) and \( \vec b \), and \( || \vec a || \) and \( || \vec b || \) are their magnitudes.
Cross Product
The cross product of vectors \( \vec a \) and \( \vec b \) is defined by:
\( \vec a \times \vec b = || \vec a || \; || \vec b || \sin \theta \; \vec n = \begin{vmatrix}
\vec i & \vec j & \vec k \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3
\end{vmatrix} \)
\( \quad \quad = \vec i \begin{vmatrix}
a_2 & a_3 \\
b_2 & b_3
\end{vmatrix}
- \vec j \begin{vmatrix}
a_1 & a_3 \\
b_1 & b_3
\end{vmatrix}
+ \vec k \begin{vmatrix}
a_1 & a_2 \\
b_1 & b_2
\end{vmatrix} \)
\( \quad \quad = (a_2 b_3 - a_3 b_2) \vec i - (a_1 b_3 - a_3 b_1) \vec j + (a_1b_2 - a_2 b_1) \vec k \)
\( \vec n \) is a unit vector perpendicular to vectors \( \vec a \) and \( \vec b \), such that \( \vec a \), \( \vec b \), and \( \vec n \) form a right-handed coordinate system.
Scalar Triple Product of Three Vectors
The scalar triple product of \( \vec a \), \( \vec b \), and \( \vec c \) is given by:
\( (\vec a \times \vec b) \cdot \vec c = \vec a \cdot (\vec b \times \vec c) \)
\( \quad \quad = \begin{vmatrix}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{vmatrix} \)
\( = a_1 \begin{vmatrix}
b_2 & b_3 \\
c_2 & c_3
\end{vmatrix} - a_2 \begin{vmatrix}
b_1 & b_3 \\
c_1 & c_3
\end{vmatrix} + a_3 \begin{vmatrix}
b_1 & b_2 \\
c_1 & c_2
\end{vmatrix} \)
\( = a_1( b_2 c_3 - b_3 c_2) - a_2 (b_1 c_3 - b_3 c_1) + a_3 (b_1 c_2 - b_2 c_1) \)
Vector Triple Product of Three Vectors
\( \vec a \times (\vec b \times \vec c) = (\vec a \cdot \vec c) \vec b - (\vec a \cdot \vec b) \vec c \)
More References and Links
Handbook of Mathematical Functions
Mathematics for Engineers with Examples and Solutions