Question

If c be a given non-zero scalar, and A and B be given non-zero vectors such that $A\perp B,$ find the vector X which satisfies the equations A. X=c and $A\times X=B$

• A. $X=\frac{cA+A\times B}{{\left|A\right|}^2}$
• B. $X=\frac{cA-A\times B}{{\left|A\right|}^2}$
• C. $X=\frac{cB-A\times B}{{\left|B\right|}^2}$
• D. $X=\frac{cB+A\times B}{{\left|B\right|}^2}$

Answer 1

The correct option is B $X=\frac{cA-A\times B}{{\left|A\right|}^2}$

We are given $A.X=c$...(1) $A\times X=B$ ...(2)
Taking cross-product with A on both sides of (2),
$A\times (A\times X)=A\times B$
i.e., $(A.X)A-(A.A)X=A\times B.$
Now using (1), we get $cA-{\left|A\right|}^{2}X=A\times B$
or $X{\left|A\right|}^2=cA-A\times B\therefore X=\frac{cA-A\times B}{{\left|A\right|}^2}.$