Question

Given $A+B=a,A\times B=b,A\cdot a=1$ Find the vector $B$.

• A. $\frac{(b\times a)+b(a^2-1)}{a^2}$
• B. $\frac{(b\times a)+b(a^2-1)}{b^2}$
• C. $\frac{(b\times a)+a(a^2-1)}{a^2}$
• D. $\frac{(b\times a)+a(a^2-1)}{b^2}$

Answer 1

Answer: (C)

$\frac{(b\times a)+a(a^2-1)}{a^2}$

Given

$A+B=a$-------(1)

$A\times B=b$------(2)

$A\cdot a=1$------(3)

cross product w.r to B

$A\times B+B\times B=a\times B$

$b=a\times B$ from eq (2)

cross product w.r to a

$b\times a=(a\times B)\times a$

$b\times a=(a\cdot a)B-(a\cdot B)a$

$b\times a+(a\cdot B)a=a^{2}B$

putting value of B from eq (1)

$b\times a+(a\cdot (a-A\left)\right)a=a^{2}B$

$b\times a+(a^2-A\cdot a)a=a^{2}B$

$b\times a+(a^2-1)a=a^{2}B$

$a^{2}B=b\times a+a(a^2-1)$

$B=\frac{b\times a+a(a^2-1)}{a^2}$