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 $\overrightarrow{X}$ of length X pointing from left to right horizontally

$\overrightarrow{X}=X\overrightarrow{x}$

A vector $\overrightarrow{Y}$ of length X pointing from bottom to top vertically

$\overrightarrow{Y}=Y\overrightarrow{y}$

From above, A vector $\overrightarrow{X}$ of length X pointing from left to right horizontally

$\overrightarrow{A}=A\overrightarrow{a}$

$A=\frac{\overrightarrow{A}}{\overrightarrow{a}}$

$\overrightarrow{a}=\frac{\overrightarrow{A}}{A}$

Addition of 2 vector

A Vector sum of 2 vectors

$\overrightarrow{Z}=\overrightarrow{X}+\overrightarrow{Y}=X\overrightarrow{x}+Y\overrightarrow{y}$

Example

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$\overrightarrow{Z}=\overrightarrow{X}+\overrightarrow{Y}=3\overrightarrow{x}+2\overrightarrow{y}$

Substraction of 2 vector

A Vector difference of 2 vectors

$\overrightarrow{Z}=\overrightarrow{X}-\overrightarrow{Y}=X\overrightarrow{x}-Y\overrightarrow{y}=X\overrightarrow{x}+(-Y\overrightarrow{y})$

Example

$\overrightarrow{Z}=\overrightarrow{X}-\overrightarrow{Y}=3\overrightarrow{x}-2\overrightarrow{y}=3\overrightarrow{x}+(-2\overrightarrow{y})$

Multiplication by a scalar

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

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

$\widehat{𝐛}=\frac{𝐛}{|𝐛|}.$