MATH 127
Vector Analysis
♦ Books: (1) Vector Analysis - Sam Series
(2) Linear Algebra - Abdur Rahman
scalar \(\rightarrow\) Magnitude
Vector \(\rightarrow\) Magnitude + Direction
\(\vec{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}\)
\(|\vec{A} = \sqrt{{a_1}^2 + {a_2}^2 + {a_3 }^3} \rightarrow \textrm{Magnitude of }~ \vec{A}\)
\(\hat{i},\hat{j},\hat{k} ~ \rightarrow \textrm{Standard basic vector } \rightarrow ~\textrm{Unit Vector}\)• Unit Vector \(\rightarrow\) Magnetitude = 1
Proof:
\begin{align*}
\hat{a} &= \frac{\vec{A}}{|\vec{A}|} = \frac{a_1}{\sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2}} \hat{i} + \frac{a_2}{\sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2}}\hat{j} + \frac{a_3}{\sqrt{{a_1 }^2 + {a_2}^2 + {a_3}^2}}\hat{k} \\
|\hat{a}| & = \sqrt{\left( \frac{a_1}{\sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2}} ~\hat{i} \right)^2 + \left( \frac{a_2}{\sqrt{{a_1}^2 + {a_2}^2 + {a_3}^2}}~\hat{j} \right)^2 + \left( \frac{a_3}{\sqrt{{a_1 }^2 + {a_2}^2 + {a_3}^2}}~\hat{k} \right)^2} \\
& \sqrt{\frac{{a_1}^2 + {a_2}^2 + {a_3}^2}{{a_1}^2 + {a_2}^2 + {a_3}^2}}\\
& = 1
\end{align*}
\begin{align*}
\vec{A} & = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \\
& = (a_1, a_2 ,a_3) ~~\rightarrow \textrm{Components}
\end{align*}
\begin{align*}
& \vec{A} = \vec{B} \implies ~|\vec{A} |=|\vec{B}| ~\textrm{and same direction} \\
& \vec{A} = - \vec{B} \implies ~|\vec{A} |=|\vec{B}| ~\textrm{but has opposite direction}
\end{align*}
• Position Vector:
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