Resolution of Vectors

A vector can be broken down into simpler parts called components that act in specific directions. Resolution of a vector means splitting one vector into two or more smaller vectors that add up to give the original vector. This technique helps us understand how a force, velocity, or any vector quantity works in different directions. Rectangular components are the most common type, where vectors are split into perpendicular directions (x and y in 2D, or x, y, and z in 3D). Breaking vectors into components makes solving physics problems much easier.

A vector \[\vec V\] can be expressed as the sum of two or more vectors along fixed directions. This process is known as vector resolution.

When a vector is resolved into components along mutually perpendicular directions (like x and y axes in 2D, or x, y, and z axes in 3D), these components are called rectangular or Cartesian components.

• Identify the vector \[\vec R\] and the angle θ it makes with the x-axis.
• Draw perpendiculars from the tip of the vector to both the x-axis and the y-axis.
• The perpendicular from the vector tip to the x-axis gives the vertical component \[\vec R_y\] at point B.
• The perpendicular from the vector tip to the y-axis gives the horizontal component \[\vec R_x\] at point A.
• Use trigonometry to calculate:
  Horizontal component: R_x = R cos θ
  Vertical component: R_y = R sin θ
• Express the vector in unit vector notation: \[\vec R\] = R_x î + R_y ĵ
• If needed, find the magnitude: R = √(Rx² + Ry²)
• If needed, find the direction: θ = tan⁻¹(Ry/Rx)

Find a unit vector in the direction of \[\vec V\] = 3 \[\hat i\] + 4 \[\hat j\].

Solution:

Magnitude |\[\vec V\]| = \[\sqrt{3^{2}+4^{2}}\] = 5.
A unit vector in the same direction is:
\[\hat u\] = ​\[\frac{\vec{V}}{|\vec{V}|}=\frac{3}{5}\hat{i}+\frac{4}{5}\hat{j}\]

Given \[\vec a\] =1 \[\hat i\] + 2 \[\hat j\] and \[\vec b\] = 2 \[\hat i\] + 1 \[\hat j\]​, what are their magnitudes? Are the vectors equal?

Solution:

\[|\vec{a}|=\sqrt{1^2+2^2}=\sqrt{5},\quad|\vec{b}|=\sqrt{2^2+1^2}=\sqrt{5}\]​

Both have the same magnitude, but their components differ:

\[a_{x}\neq b_{x},a_{y}\neq b_{y}.\]

Thus \[\vec a\] ≠ \[\vec b\].