Lecture 1 [04 Feb, 2020] MATH 127 (Maj Sultana)

MATH 127 

Vector Analysis

    ♦ Books: (1) Vector Analysis - Sam Series
                  (2) Linear Algebra - Abdur Rahman

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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: