Resolution of Vectors – Class 11 | Chapter – 4 | Physics Short Notes Series PDF for NEET & JEE
Resolution of Vectors: The resolution of vectors refers to the process of breaking down a vector into its components along two or more perpendicular directions. The components of a vector are the projections of the vector onto these directions, and they can be added or subtracted to obtain the original vector.
Types of Resolution of Vectors
There are two common types of resolution of vectors in physics: rectangular or Cartesian resolution and polar resolution.
- Rectangular or Cartesian Resolution: In rectangular resolution, a vector is resolved into two or three components along the x, y, and z axes of a Cartesian coordinate system. The x, y, and z components of a vector are obtained by taking the projections of the vector onto the corresponding axes. For example, if v is a vector in three dimensions, then its x-component, vx, is given by the projection of v onto the x-axis. Similarly, the y-component, vy, is given by the projection of v onto the y-axis, and the z-component, vz, is given by the projection of v onto the z-axis.
- Polar Resolution: In polar resolution, a vector is resolved into two components along two perpendicular directions, usually denoted as r and θ. The r-component of the vector is the magnitude of the vector, while the θ-component is the angle that the vector makes with respect to a reference direction. Polar resolution is often used when dealing with circular motion, where the r-component represents the radial distance from the center of the circle, and the θ-component represents the angle of rotation.
Components of Resolution of Vectors
To find the components of a vector, we typically use trigonometry and the Pythagorean theorem. Here are the steps involved in resolving a vector into its components:
- Choose a set of perpendicular axes, usually denoted by x, y, and z.
- Determine the angle that the vector makes with each axis, usually denoted by θ, φ, and ψ.
- Use trigonometry to find the projections of the vector along each axis. For example, if the vector makes an angle θ with the x-axis and has a magnitude of |v|, then its x-component, vx, is given by vx = |v|cos(θ).
- Repeat this process for each axis to find the components of the vector along each direction.
In two dimensions, a vector can be resolved into two components, usually denoted by vx and vy. In three dimensions, a vector can be resolved into three components, usually denoted by vx, vy, and vz.
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