PX153 - A4 - vector operations

addition and subtraction

• properties
  • commutative - $\overrightarrow{u}\pm \overrightarrow{v}=\overrightarrow{v}\pm \overrightarrow{u}$
  • associative - $\overrightarrow{u}\pm (\overrightarrow{v}\pm \overrightarrow{w})=(\overrightarrow{v}\pm \overrightarrow{u})\pm \overrightarrow{w}$
    
    

vector products

scalar/dot product

• $s=\overrightarrow{u}\cdot \overrightarrow{v}$
  • $s$ is a scalar
• geometrically, it is the magnitude of one vector projected on the other

- $\theta =$ angle between the vectors

• if $\theta =\frac{\pi }{2}$ (orthogonal vectors), $s=0$
• for cartesian coordinate basis vectors

and
$$\underline{\hat i}\cdot \underline{\hat j}=\underline{\hat j}\cdot \underline{\hat k}=\underline{\hat k}\cdot \underline{\hat i}=0$$
- this denotes that the basis vectors are orthonormal

• properties
  • commutative - $\overrightarrow{u}\cdot \overrightarrow{v}=\overrightarrow{v}\cdot \overrightarrow{u}$
  • distributive - $\overrightarrow{u}\cdot (\overrightarrow{v}+\overrightarrow{w})=\overrightarrow{v}\cdot (\overrightarrow{u}+\overrightarrow{w})$
• in cartesian coordinates - $s=\overrightarrow{u}\cdot \overrightarrow{v}=u_x{v}_x+u_y{v}_y+u_z{v}_z$
• using both definitions:$$\cos \theta =\frac{\overrightarrow{u}\cdot \overrightarrow{v}}{|u||v|}$$
  • component of $\overrightarrow{u}$ along $\overrightarrow{v}$ $={\overrightarrow{u}}_v=(\frac{\overrightarrow{u}\cdot \overrightarrow{v}}{v^2})\overrightarrow{v}=(\frac{\overrightarrow{u}\cdot \overrightarrow{v}}{v})\underset{―}{\hat{v}}$
• eg of scalar product in physics:$$W=\overrightarrow{F}\cdot \overrightarrow{d}$$

vector/cross product

• $\overrightarrow{v}=\overrightarrow{u}\times \overrightarrow{w}$

  • $\overrightarrow{v}$ is a vector

• geometrically, $\overrightarrow{u}$ and $\overrightarrow{w}$ define a parallelepiped of area $uw\sin \theta$
  

• $\overrightarrow{v}$ is a vector perpendicular to the plane of $\overrightarrow{u}$ and $\overrightarrow{w}$

  • magnitude $uw\sin \theta$
  • direction is set by the right-hand rule $$\vec v = uw\sin{\theta}\underline{\hat e}_v$$
  • $\overrightarrow{v}$ is the same independent of the coordinate system, its component in a given basis changes

• cases

  • if $\overrightarrow{u}$ and $\overrightarrow{w}$ are parallel, $\theta =0$ and $\overrightarrow{u}\times \overrightarrow{w}=0$
  • if $\overrightarrow{u}$ and $\overrightarrow{w}$ are not parallel, $\overrightarrow{u}\cdot (\overrightarrow{u}\times \overrightarrow{w})=0$

• for cartesian coordinates,

  • $\underset{―}{\hat{i}}\times \underset{―}{\hat{i}}=\underset{―}{\hat{j}}\times \underset{―}{\hat{j}}=\underset{―}{\hat{k}}\times \underset{―}{\hat{k}}=0$
  • $\underset{―}{\hat{i}}\times \underset{―}{\hat{j}}=\underset{―}{\hat{k}}$ , $\underset{―}{\hat{j}}\times \underset{―}{\hat{k}}=\underset{―}{\hat{i}}$ , $\underset{―}{\hat{k}}\times \underset{―}{\hat{i}}=\underset{―}{\hat{j}}$

• properties:

  • vector product is NOT commutative, ie: $\underset{―}{\hat{i}}\times \underset{―}{\hat{j}}\ne \underset{―}{\hat{j}}\times \underset{―}{\hat{i}}$
  • it is anti-commutative, ie: $\underset{―}{\hat{j}}\times \underset{―}{\hat{i}}=-\underset{―}{\hat{k}}$

• calculating cross product:$$\vec v = \vec u \times \vec w = (u_yw_z-u_zw_y)\underline{\hat i}+(\vec u_zw_x-u_xw_z)\underline{\hat j}+(\vec u_xw_y-u_yw_x)\underline{\hat k}$$

  • easy tip:
    or,
    
    

• many physical quantities are defined by cross product
  - eg: $\overrightarrow{L}=\overrightarrow{r}\times (m\overrightarrow{v})$, angular momemtum

• eg of cross product calculation