Calculus II/Vector Algebra Operation

Definitions of a vector
A vector is a mathematical concept that has both magnitude and direction. A vector is an object that has certain properties:
 * Magnitude (or length) denoted as A
 * Direction . denoted as a

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A vector $$\vec X$$ of length X pointing from left to right horizontally
 * $$\vec X = X \vec x $$

A vector $$\vec Y$$ of length X pointing from bottom to top vertically
 * $$\vec Y = Y \vec y$$

From above, A vector $$\vec X$$ of length X pointing from left to right horizontally
 * $$\vec A = A \vec a $$
 * $$ A = \frac{\vec A}{\vec a} $$
 * $$ \vec a = \frac{\vec A}{A} $$

Addition of 2 vector
A Vector sum of 2 vectors
 * $$\vec Z =  \vec X + \vec Y = X  \vec x + Y  \vec y$$

Example

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 * $$\vec Z =  \vec X + \vec Y = 3  \vec x + 2  \vec y$$

Substraction of 2 vector
A Vector difference of 2 vectors
 * $$\vec Z =  \vec X - \vec Y = X  \vec x - Y  \vec y =  X  \vec x + (- Y  \vec y)$$

Example
 * $$\vec Z =  \vec X - \vec Y = 3  \vec x - 2  \vec y = 3  \vec x + (- 2  \vec y) $$

Multiplication by a scalar
Multiplication of a vector $$\mathbf{b}\,$$ by a scalar $$\lambda\,$$ has the effect of stretching or shrinking the vector (see Figure 2(b)).

You can form a unit vector $${\mathbf{\hat b}}\,$$ that is parallel to $$\mathbf{b}\,$$ by dividing by the length of the vector $$|\mathbf{b}|\,$$. Thus,

{\mathbf{\hat b}} = \cfrac{\mathbf{b}}{|\mathbf{b}|} ~. $$