Vector Calculus Questions

Question Number 1016 by 123456 last updated on 14/May/15

$$\left[\left(\overset{\rightarrow} {{x}}×\overset{\rightarrow} {{a}}\right)×\overset{\rightarrow} {{b}}\right]×\overset{\rightarrow} {{x}}=\overset{\rightarrow} {{c}} \\$$

Answered by rpatle69@gmail.com last updated on 14/May/15

$$\left[−\overset{\rightarrow} {{b}}×\left(\overset{\rightarrow} {{x}}\:×\overset{\rightarrow} {{a}}\right)\right]×\overset{\rightarrow} {{x}}=\overset{\rightarrow} {{c}} \\$$$$\overset{\rightarrow} {{x}}×\left[\overset{\rightarrow} {{b}}×\left(\overset{\rightarrow} {{x}}×\overset{\rightarrow} {{a}}\right)\right]=\overset{\rightarrow} {{c}} \\$$$$\overset{\rightarrow} {{x}}×\left[\left(\overset{\rightarrow} {{b}}.\overset{\rightarrow} {{a}}\right)\overset{\rightarrow} {{x}}−\left(\overset{\rightarrow} {{b}}.\overset{\rightarrow} {{x}}\right)\overset{\rightarrow} {{a}}\right]=\overset{\rightarrow} {{c}} \\$$$$\\$$$$\\$$

Commented by prakash jain last updated on 17/May/15

$$\overset{\rightarrow} {{x}}×\left(\overset{\rightarrow} {{b}}\centerdot\overset{\rightarrow} {{a}}\right)\overset{\rightarrow} {{x}}−\left(\overset{\rightarrow} {{b}}.\overset{\rightarrow} {{x}}\right)\left(\overset{\rightarrow} {{x}}×\overset{\rightarrow} {{a}}\right)=−\left(\overset{\rightarrow} {{b}}\centerdot\overset{\rightarrow} {{x}}\right)\left(\overset{\rightarrow} {{x}}×\overset{\rightarrow} {{a}}\right) \\$$

Commented by 123456 last updated on 17/May/15

$$−\left(\overset{\rightarrow} {{b}}\centerdot\overset{\rightarrow} {{x}}\right)\left(\overset{\rightarrow} {{x}}×\overset{\rightarrow} {{a}}\right)\centerdot\overset{\rightarrow} {{a}}=\mathrm{0}=\overset{\rightarrow} {{c}}\centerdot\overset{\rightarrow} {{a}} \\$$$$−\left(\overset{\rightarrow} {{b}}\centerdot\overset{\rightarrow} {{x}}\right)\left(\overset{\rightarrow} {{x}}×\overset{\rightarrow} {{a}}\right)\centerdot\overset{\rightarrow} {{x}}=\mathrm{0}=\overset{\rightarrow} {{c}}\centerdot\overset{\rightarrow} {{x}} \\$$

Commented by 123456 last updated on 17/May/15

$$\overset{\rightarrow} {{x}}=\lambda_{\mathrm{1}} \overset{\rightarrow} {{a}}+\lambda_{\mathrm{2}} \left(\overset{\rightarrow} {{a}}×\overset{\rightarrow} {{c}}\right) \\$$$$\mathrm{for}\:\mathrm{some}\:\lambda_{\mathrm{1}} ,\lambda_{\mathrm{2}} \in\mathbb{R} \\$$