Question

the resultant of vectors of vector a and vector b is perpendicular to vector a also the magnitude of vector A is equals to magnitude of vector C the angle between a and b is

Answer 1

Let,R be the resultant of the sum of the two vectors A⃗ and B⃗ such that |A⃗ |=|B⃗ |.So,
R=|A⃗ |2+|B⃗ |2+2|A⃗ ||B⃗ |cosθ−−−−−−−−−−−−−−−−−−−−√
where θ is the angle between A⃗ and B⃗ .So,

R=2|A⃗ |2+2|A⃗ |2cosθ−−−−−−−−−−−−−−√
.

Now magnitude of resultant R is equal to either of the vectors A⃗ and B⃗ .So,

|A|=2|A⃗ |2+2|A⃗ |2cosθ−−−−−−−−−−−−−−√
⟹|A⃗ |2=2|A⃗ |2+2|A⃗ |2cosθ
⟹−|A⃗ |22|A⃗ |2=cosθ
⟹θ=cos−1−12


Hope this helps!!