# Vector Operations

## Key Questions

• Vectors can be added by adding the components individually as long as they have the same dimensions. Adding two vectors simply gives you a resultant vector.

What that resultant vector means depends on what quantity the vector represents. If you are adding a velocity with a change of velocity, then you would get your new velocity. If you are adding 2 forces, then you would get a net force.

If you are adding two vectors that have the same magnitude but opposite directions, your resultant vector would be zero. If you are adding two vectors that are in the same direction, then the result is in the same direction with a magnitude that is the sum of the 2 magnitudes.

• Given two vectors $\vec{v}$ and $\vec{w}$ you have:
$\vec{v} - \vec{w} = \vec{v} + \left(- \vec{w}\right)$

Graphically we can use the Parallelogram Law:

If you have the vectors in components form you again use:
$\vec{v} - \vec{w} = \vec{v} + \left(- \vec{w}\right)$ operating on each set of corresponding components.

For example:
$\vec{v} = 4 \vec{i} + 2 \vec{j} - 5 \vec{k}$ and:
$\vec{w} = - 2 \vec{i} + 4 \vec{j} + \vec{k}$

$\vec{v} - \vec{w} = \vec{v} + \left(- \vec{w}\right) = \left[4 + \left(2\right)\right] \vec{i} + \left[2 + \left(- 4\right)\right] \vec{j} + \left[- 5 + \left(- 1\right)\right] \vec{k} =$
$= 6 \vec{i} - 2 \vec{j} - 6 \vec{k}$

• The formula for the vertical component of a vector ai + bj is as follows:

v_y = ||A|| sin(θ)

First, calculate the magnitude of the vector A which is $| | A | |$:
||A|| = $\sqrt{{a}^{2} + {b}^{2}}$

Next, determine $\theta$
If you draw a triangle where a is the x axis and b is the y axis, you get a right triangle. The angle $\theta$ has the following measurement below:
$\tan \left(\theta\right) = \frac{b}{a}$
$\theta = a r t c a n \left(\frac{b}{a}\right)$

Finally, we have the vertical component formula:
v_y = ||A|| sin(θ)

For calculator assistance, use the component button here:

http://www.mathcelebrity.com/vector.php

• Consider a vector $\vec{v}$, for example, in space:

If you want to describe it to, say, a friend you can say that has a "modulus" (=length) and direction (you may use, for example, North, South, East, west...etc.).

There is also another way to describe this vector.
You must take your vector into a reference frame to have some numbers related to it and then you take the coordinates of the tip of the arrow...your COMPONENTS !
You can now write your vector as: $\vec{v} = \left(a , b\right)$

For Example: $\vec{v} = \left(6 , 4\right)$

In 3 dimensions you simply add a third component on the $z$ axis.
For example: $\vec{w} = \left(3 , 5 , 4\right)$