12.2 Vectors

Vectors are quantities that have a magnitude (size) and direction (length). Written as $\boldsymbol{v}$ or $\vec{v}$.

Two vectors are equivalent if they have the same magnitude and direction. $\vec{u}=\vec{AB}$ $\vec{v}=\vec{CD}$ $\vec{u}=\vec{v}$ despite being at different locations.

A zero vector, $\vec{0}$, has length of $0$ and no direction.

Combining Vectors

To add the vectors $\vec{u}$ with $\vec{v}$, position the initial point of $\vec{v}$ at the terminal point of $\vec{u}$. The sum, $\vec{u+v}$, would be the vector created from the initial point of $\vec{u}$ to the terminal point of $\vec{v}$. This is called the "Triangle Law." The order does not matter.

Negative scalars change the direction.

Multiplying a vector by $c$ changes its length. Real numbers work as a scaling factor which is why they are called "scalars". Non-zero vectors are parallel if they are scalar multiples. $\vec{b}$ - $\vec{a}$ would be the same as $\vec{b}+(-\vec{a})$. Flip $\vec{a}$ and then apply the triangle law.

Components

Sometimes we place vectors at the origin. Components are the individual $x$/$y$/$z$ movements. $\vec{a}=<a_{1},a_{2},a_{3}>$ means move $a_{1}$ units in the $x$ direction, $a_{2}$ units in the $y$ direction, and $a_{3}$ units in the $z$ direction. Vectors that start at the origin are called "position vectors".

To find the vector, $\vec{a}$, between two points, $A(x_{1},y_{1},z_{1})$ and $B(x_{2},y_{2},z_{2})$: $\vec{a}=<x_{2}-x_{1},y_{2}-y_{1},z_{2}-z_{1}>$ $final-initial$

Magnitude of Vector

The magnitude or length of vector $\vec{v}$ is denoted by $|\vec{v}|$ or $||\vec{v}||$. The length of a 2D vector $\vec{a}=<a_{1}, a_{2}>$ is $|\vec{a}|=\sqrt{a_{1}^2+a_{2}^2}$. The length of a 3D vector $\vec{a}=<a_{1}, a_{2}, a_{3}>$ is $|\vec{a}|=\sqrt{a_{1}^2+a_{2}^2+a_{3}^2}$. This is essentially the distance from the point to the origin.

Manipulating Vectors

Adding

To add vectors algebraically, add the components. Given $\vec{a}=<a_{1},a_{2}>$ and $\vec{b}=<b_{1},b_{2}>$, $\vec{a}+\vec{b}=<a_{1}+b_{1},a_{2}+b_{2}>$

Multiply by Scalar

To multiply vectors by a scalar, multiply each of the components. Given $\vec{a}=<a_{1},a_{2}>$ and scalar $c$, $c\vec{a}=<ca_{1},ca_{2}>$

Properties of vectors

• $\vec{a}+\vec{b} = \vec{b}+\vec{a}$
• $\vec{a}+\vec{0}=\vec{a}$
• $\vec{a}+(\vec{b}+\vec{c}) = (\vec{a}+\vec{b})+\vec{c}$
• $\vec{a}+(-\vec{a}) = \vec{0}$
• $c(\vec{a}+\vec{b}) = c\vec{a}+c\vec{b}$
• $(cd)\vec{a} = c(d\vec{a})$
• $(c+d)\vec{a} = c\vec{a}+d\vec{a}$
• $1\vec{a}=\vec{a}$

Special vectors

$\hat{i}=<1,0,0>$ $\hat{j}=<0,1,0>$ $\hat{k}=<0,0,1>$

The following forms are interchangable. Try to use the same format as in the prompt. $\vec{a}=<a_{1}, a_{2}, a_{3}>=a_{1}\hat{i}+a_{2}\hat{j}+a_{3}\hat{k}$

Unit Vector

A unit vector is a vector whose length is 1. $\hat{i}$, $\hat{j}$, $\hat{k}$ are all unit vectors. Turn a vector $\vec{a}$ into a unit vector by dividing the vector by its magnitude. $\vec{u}=\frac{1}{|\vec{a}|}\vec{a}=\frac{\vec{a}}{|\vec{a}|}$ Useful for unit spheres.