Question

Prove that there is only one and one way in which a given vector can be resolved in given directions.

Answer 1

Hint: Vector can be defined as a physical quantity that has both magnitude and direction. Also, a vector is a physical quantity that can be represented as a straight line with an arrow head. The formula used for proving the statement is given below.

Formula used:
The formula used for calculating the force acting on the horizontal vector is given below
$F_x=F\cos \theta$
Here, $F_x$ is the force acting on the horizontal component, $F$ is the force on vectors and $\theta$ is the angle between the vectors.
Also, the formula used for calculating the force acting on the vertical vector is given below
$F_y=F\sin \theta$
Here, $F_y$ is the force acting on the vertical component, $F$ is the force on vectors and $\theta$ is the angle between the vectors.

Complete step by step answer:
Let us consider a vector $A$. Let this vector have two components from which one will be along the horizontal axis and the other will be along the vertical axis. Now, the formula used for calculating the force acting on the horizontal vector is given below
$F_x=F\cos \theta$
Also, the formula used for calculating the force acting on the vertical vector is given below
$F_y=F\sin \theta$
Now, we can see that the direction of both the vectors can be calculated using the trigonometric function. Therefore, we can say that there is only one and one way in which a given vector can be resolved in given directions.Hence proved.

Note:Here, in the above question, the direction of the horizontal and vertical vectors can be calculated by using the cosine and sine of the angle between the vectors. Since both the cosine and sine are the trigonometric functions. That is why the method of resolving direction is the same.