a and b are mutually perpendicular unit vectors. If r is a vector satisfying r ⋅ a=0, r ⋅ b=1 and [r a b] = 1 then r is

Question

a and b are mutually perpendicular unit vectors. If r is a vector satisfying r ⋅ a=0, r ⋅ b=1 and [r a b] = 1 then r is (A) a × b+b (B) a+(a × b) (C) b+(a × b) (D) a × b+a

Tardigrade Admin · Accepted Answer

Let r=x1 a+x2 b+x3(a × b) Then, r ⋅ a=x1|a|2, r ⋅ b=x2|b|2 and, r ⋅(a × b)=x3|a × b|2 ⇒ x1=0, x2=1, x3=1 ∴ r=a × b+b

Q. $a$ and $b$ are mutually perpendicular unit vectors. If $r$ is a vector satisfying $r \cdot a = 0, r \cdot b = 1$ and $[r a b] = 1$ then $r$ is

$a \times b + b$

$a + (a \times b)$

$b + (a \times b)$

$a \times b + a$

Let $r = x_{1} a + x_{2} b + x_{3} (a \times b)$
Then, $r \cdot a = x_{1} ∣ a ∣^{2}, r \cdot b = x_{2} ∣ b ∣^{2}$
and, $r \cdot (a \times b) = x_{3} ∣ a \times b ∣^{2}$
$\Rightarrow x_{1} = 0, x_{2} = 1, x_{3} = 1$
$∴ r = a \times b + b$

Q. a and b are mutually perpendicular unit vectors. If r is a vector satisfying r⋅a=0,r⋅b=1 and [rab]=1 then r is

a×b+b

a+(a×b)

b+(a×b)

a×b+a