1. Scalars vs. Vectors
- scalar
-
number (with units)
- vector
-
magnitude (a number with units) and a direction
2. Unit Vectors in the Rectangular Coordinate System
- Tougaw’s convention
-
-
\(\mathbf{a}_x\) (bold lowercase “a”)
-
\(\hat{a}_x\) (“a hat”, hat only used over unit vectors)
-
3. Vector Magnitude
The length of the vector along its direction.
Obtain a unit vector in the direction of a vector \(\vec{A}\) by dividing it by its own magnitude.
\[\mathbf{a}_A = \frac{\vec{A}}{\left|\vec{A}\right|}\]
5. Dot Products
Definition:
\[\mathbf{\vec{A}} \bullet \mathbf{\vec{B}} = \left|A\right| \cdot \left|B\right| \cdot \cos{\theta}\]
Dot product takes two and yields a .
Fill in those blanks first, then click for the answer
In rectangular coordinates:
\[\mathbf{\vec{A}} \bullet \mathbf{\vec{B}} = A_x B_x + A_y B_y + A_z B_z\]
a.k.a the inner product:
\[\begin{bmatrix}
A_x & A_y & A_z
\end{bmatrix}
\cdot
\begin{bmatrix}
B_x \\ B_y \\ B_z
\end{bmatrix}\]
7. Cross Products
Cross product takes two and yields a .
Fill in those blanks first, then click for the answer
Magnitude:
\[\left|\mathbf{\vec{A}} \times \mathbf{\vec{B}}\right| = \left|A\right| \cdot \left|B\right| \cdot \sin{\theta}\]
Direction is orthogonal to both vectors, right-hand rule for the direction.