What do you mean by resolution of vectors? Explain how a vector can be resolved into its rectangular components in a plane?

Resolution of vectors refers to the process of breaking down a vector into its individual components along specified directions, typically perpendicular axes. In a two-dimensional plane, vectors can be resolved into their rectangular components using trigonometric principles. This process is essential in physics and engineering, as it allows us to analyze and manipulate vectors more easily.

Here's how a vector can be resolved into its rectangular components in a plane:

Define Coordinate Axes: Choose two perpendicular axes in the plane. Conventionally, these axes are labeled as the x-axis and the y-axis. The x-axis typically represents the horizontal direction, and the y-axis represents the vertical direction.

Determine the Vector: Identify the vector you want to resolve. This vector can be represented by an arrow with a specific magnitude (length) and direction in the plane.

Break Down the Vector: To resolve the vector into its components, you need to determine the projections of the vector onto the x-axis and y-axis. These projections will be the magnitudes of the vector's components along those axes.

a. Find the horizontal component (x-component):

Use trigonometry (usually cosine) to find the magnitude of the projection of the vector onto the x-axis. This is done by multiplying the magnitude of the vector by the cosine of the angle θ between the vector and the positive x-axis.
Mathematically, the x-component (Vx) is given by Vx = V * cos(θ), where V is the magnitude of the vector.
b. Find the vertical component (y-component):

Similarly, use trigonometry (usually sine) to find the magnitude of the projection of the vector onto the y-axis. This is done by multiplying the magnitude of the vector by the sine of the angle θ between the vector and the positive x-axis.
Mathematically, the y-component (Vy) is given by Vy = V * sin(θ), where V is the magnitude of the vector.
Resultant Vector: Once you've found the x and y components, you can represent the original vector as the sum of these components. The resultant vector is obtained by combining the x and y components using vector addition. The direction of the resultant vector is typically given by the angle θ, and its magnitude is the same as the magnitude of the original vector.

Mathematically, the resultant vector (V) can be expressed as V = √(Vx^2 + Vy^2), and the angle θ is given by θ = arctan(Vy / Vx).

By resolving vectors into their rectangular components, you can simplify vector operations and analyze their effects on different axes independently, which is particularly useful in physics problems involving forces, motion, and other vector quantities.