ElectroMagnetic Theory For GATE/IES
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Properties of Vector Operations:
\begin{array}{llll}{\text { 1. }} & {\vec{a}+\vec{b}=\vec{b}+\vec{a}} & {\text { 2. }} & {\vec{a}+(\vec{b}+\vec{c})=(\vec{a}+\vec{b})+\vec{c}} \\ {\text { 3. }} & {\vec{a}+\overrightarrow{0}=\vec{a}} & {\text { 4. }} & {\vec{a}+(-\vec{a})=\overrightarrow{0}} \\ {\text { 5. }} & {c(\vec{a}+\vec{b})=c \vec{a}+c \vec{b}} & {\text { 6. }} & {(c+d) \vec{a}=c \vec{a}+d \vec{a}} \\ {7 .} & {(c d) \vec{a}=c(d \vec{a})} & {\text { 8. }} & {1 \vec{a}=\vec{a}}\end{array}
Scalar product:
\begin{array}{lll}{\text { 1. }} & {\vec{a} \cdot \vec{a}=|\vec{a}|^{2}} & {\text { 2. } \quad \vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{a}} \\ {\text { 3. }} & {\vec{a} \cdot(\vec{b}+\vec{c})=\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c}} & {(c \vec{a}) \cdot \vec{b}=c(\vec{a} \cdot \vec{b})} \\ {5 .} & {\overrightarrow{0} \cdot \vec{a}=0} & {\text { 6. }} & {\vec{a} \cdot \vec{b}=|\vec{a}||\vec{b}| \cos \theta}\end{array}
Vector product:
\begin{array}{l}{\text { 1. } \vec{a} \times \vec{b} \perp \vec{a}, \vec{b}} \\ {\text { 2. }|\vec{a} \times \vec{b}|=|\vec{a}||\vec{b}| \sin \theta} \\ {\text { 3. } \hat{i} \times \hat{y}=\hat{k}, \hat{y} \times \hat{k}=\hat{i}, \hat{k} \times \hat{i}=\hat{\jmath}} \\ {\text { 4. } \quad \vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \sin \theta \hat{n}} \\ {\text { 5. } \quad \vec{a} \times \vec{b}=0 \Leftrightarrow \vec{b} \times \vec{a}} \\ {\text { 4. } \quad(c \vec{a}) \times \vec{b}=\vec{a} \times(\vec{c} \vec{b})=c(\vec{a} \times \vec{b})} \\ {\text { 8. } \vec{a} \times(\vec{b}+\vec{c})=\vec{a} \times \vec{b}+\vec{a} \times \vec{c}} \\ {\text { 9. } \vec{a} \cdot(\vec{b} \times \vec{c})=(\vec{a} \times \vec{b}) \cdot \vec{c}} \\ {\text { 10. } \vec{a} \times(\vec{b} \times \vec{c})=(\vec{c} \cdot \vec{a}) \vec{b}-(\vec{b} \cdot \vec{a}) \vec{c}}\end{array}
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