Question

The resultant of two vectors of magnitude $2$ and $3$ is $1$ the angle between them is?

Answer 1

Here we have given the resultant of two vectors and the vectors are also known. Now by applying the resultant vector formula with the parallelogram law and by drawing a rough diagram we can find the value of angle. The angle here is the angle between the two given vectors.

Complete step by step answer:
As per the problem we have given two vectors of magnitude $2$ and $3$, the resultant of the two vectors is $1$.

From the parallelogram law of two vectors having some magnitude will give a result vectors which can be represented as,
$R = \sqrt {{{\left( A \right)}^2} + {{\left( B \right)}^2} + 2AB\cos \theta }$
Where,
$A = 2$ and $B = 3$ are two vectors which are at an angle of $\theta$.
$R$ is the resultant of two vectors which is equal to $1$.
Now putting the respective values in the above formula we will get,
$1 = \sqrt {{{\left( 2 \right)}^2} + {{\left( 3 \right)}^2} + 2 \times 2 \times 3\cos \theta }$
Taking the root over to other side and opening the brackets we will get,
$1 = 4 + 9 + 12\cos \theta$
$\Rightarrow - 12 = 12\cos \theta$
Cancelling the common terms we will get,
$- 1 = \cos \theta \Rightarrow \theta = {\cos ^{ - 1}}\left( { - 1} \right)$
Hence $\theta = 180^\circ$ as we know $\cos 180 = - 1$
Therefore the angle between the two given vectors is $\theta = 180^\circ$.
That is they are totally opposite to each other or we can say their directions are opposite.

Note: Parallelogram Law states that the resultant of two vector quantities can be represented by its magnitude , direction and sense by two adjacent sides of a parallel both of which are either directed toward or away from the point of their intersection is the diagonal of the parallelogram through that point.