Motion in a Plane | Class 11 Physics Notes

3.5 Resolution of Vectors

Resolution of vectors is the process of expressing a vector as the sum of two or more component vectors along specified directions. Given two non-collinear vectors a and b in a plane, any vector A in that plane can be expressed as A = λa + μb, where λ and μ are real numbers.

Graphically, to resolve vector A, draw lines through the tail and head of A parallel to vectors a and b, respectively. The intersection point Q defines the components OQ = λa and QP = μb.

Unit vectors are vectors of magnitude one used to specify directions without dimensions. The standard unit vectors along the x-, y-, and z-axes are denoted by î, ĵ, and k̂, respectively, and are mutually perpendicular.

Any vector A in the x-y plane can be resolved into components along î and ĵ: A = A_x î + A_y ĵ, where A_x and A_y are scalars representing the components along x and y axes.

Using trigonometry, if A makes an angle θ with the x-axis, then A_x = A cos θ and A_y = A sin θ. Conversely, given components A_x and A_y, the magnitude and direction of A can be found using Pythagoras and inverse tangent functions.

This method extends to three dimensions, where A = A_x î + A_y ĵ + A_z k̂, with components related to angles α, β, γ between A and the coordinate axes.

📊 Diagram: Fig. 3.8 (a) Two non-colinear vectors a and b; (b) Resolving vector A in terms of vectors a and b; Fig. 3.9 (a) Unit vectors î, ĵ, k̂ along x, y, z axes; (b) Vector A resolved into components A_x and A_y; (c) Components A_1 and A_2 expressed in terms of î and ĵ; (d) Vector A resolved into components along x, y, z axes.

🧪 Activity: Activity: Resolve a given vector graphically into components along two perpendicular directions and verify using trigonometric calculations.

🔗 Connection: This section leads to the analytical method of vector addition, where components are algebraically added.