All my math notes, now in Markdown.

You should be able to

- Write 2D vectors and find unit vectors and magnitudes
- Perform operations of vectors
- Solve applications including force and direction of objects

A **two dimensional vector** is an ordered pair of real numbers, denoted in component form as $\vec{v} = \langle a, b\rangle$. The numbers $a$ and $b$ and the components. If an arrow has initial point $A(x_1, y_1)$ and terminal point $(x_2, y_2)$, it represents the vector in component form $\overrightarrow{AB} = \langle x_2 - x_1, y_2 - y_1 \rangle$

The magnitude of $\vec{v}$ is the length of the arrow and the direction of $\vec{v}$ is the direction in which the arrow is pointing.

The **magnitude of vector** $\vec{v} = \langle v_1, v_2 \rangle$, represented by $\overrightarrow{AB}$ is
\(\lvert v\rvert = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(v_1)^2 + (v_2)^2}\)

The **unit vector** $\vec{u}$ is a vector with length $\lvert u\rvert = 1$.

Two component unit vectors are represented by ($\hat{\imath}$ hat and $\hat{\jmath}$ hat). $\hat{\imath} = \langle 1, 0 \rangle$ and $\hat{\jmath} = \langle 0, 1 \rangle$. The linear combination of a vector $\vec{v}$ is \(\vec{v} = \langle a, b \rangle = ai + bj\)

The sum or difference of vectors $\vec{a} \pm \vec{b}$ is $\vec{a} \pm \vec{b} = \langle a_1 \pm b_1, a_2 \pm b_2 \rangle$

The product of the scalar $k$ and the vector $\vec{v}$ is $k\vec{a} = \langle ka_1, ka_2 \rangle$

If $\vec{v}$ has direction angle $\theta$, the components of $\vec{v}$ can be computed $\vec{v} = \langle \lvert v\rvert\cos{\theta}, \lvert v\rvert\;\sin{\theta} \rangle$ and the unit vector in the direction of $\vec{v}$ is $\vec{u} = \langle \cos{\theta}, \sin{\theta} \rangle$

Weight force, $F_{g}$ directly downward, can be calculated by multiplying the mass by the acceleration.

SI units: $Newtons = kg * 9.8m/s^2$

American units: $Pounds = slugs * 32ft/s^2$

no calc. unless specified

- Given $A=(2,-3)$ and $B(5,1)$,
- Find the component form of $\overrightarrow{AB}$
- Find the mangitude of $\overrightarrow{AB}$
- Find the unit vector of $\overrightarrow{AB}$
- Write $\overrightarrow{AB}$ as a linear combination.
- Find the direction of $\overrightarrow{AB}$ (calculator)
- Write $\overrightarrow{AB}$ in its trigonometric form

- If $\vec u = \langle4,-3\rangle,\vec v=\langle1,2\rangle$
- Find $\vec w=2\vec u-2\vec v$
- Write the linear combination of $\vec w$.
- Find $\lvert\vec w\rvert$

- Given the vector $\vec v$ below,
- Write the exact component form.
- Write the exact linear combination form.

- For $\vec v = 7(\cos 135°\hat\imath + \sin135°\hat\jmath)$,
- Write in component form
- Find the magnitude of $\vec v$

- Find a vector $\vec v$ that has a magnitude of $\lvert\vec v\rvert = 3$ in the direction of $\langle-5,12\rangle$
- An airplane is flying on a compass bearning of 340° at 325 mph. A wind is blowing with the bearing of 320° at 40 mph.
- Find the component form for the velocity of the airplane and wind.
- Find the ground speed of the resultant.
- Find the new bearing of the plane.

- The minimum force required to hold up a box on a 20° frictionless inclined plane is 30lbs. What is the weight of the box?
- A force of 50lbs acts on an object at an angle of 45°. A second force of 75 lbs acts on the object at an angle of -30°. Find the magnitude of the resultant force and the direction.