Resolution
of vectors and rectangular components
A vector
directed at an angle with the co-ordinate axis, can be resolved into its
components along the axes. This process of splitting a vector into its
components is known as resolution of a vector.
Consider a vector R = Vector( OA) making an angle θ with X - axis. The vector R can be resolved
into two components along X - axis and Y-axis respectively. Draw two
perpendiculars from A to X and Y axes respectively. The intercepts on these
axes are called the scalar components Rx
and Ry.
Then, OP is Rx,
which is the magnitude of x component of Vector R and OQ is Ry, which is the magnitude of y component of Vector R .
From ∆ OPA,
cos θ = OP/OA = Rx/R
(or) Rx=Rcos θ
sin θ = OQ/OA = Ry/R
(or) Ry=Rsin θ
R2 = Rx2 + Ry2
Also, Vector R can be expressed as Vector R = Rxi + Ryj
where i and j are unit vectors.
In terms of Rx and Ry , θ can be expressed as
θ = tan−1 [Ry/Rx]
Scalar and vector quantities
A study of motion will involve the introduction
of a variety of quantities, which are used to describe the physical world.
Examples of such quantities are distance, displacement, speed, velocity,
acceleration, mass, momentum, energy, work, power etc. All these quantities can
be divided into two categories ? scalars
and vectors.
The scalar quantities have magnitude only. It is denoted by a number and unit. Examples : length, mass, time, speed, work,
energy,
temperature etc. Scalars of the same kind can
be added, subtracted, multiplied or divided by ordinary laws.
The vector quantities have both magnitude and direction. Examples:
displacement, velocity, acceleration, force, weight, momentum, etc.
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