Vectors Problem Example 3

Description:

î and ĵ are unit vectors along x and y axis respectively.

1. What is the magnitude and direction of î + ĵ , and î − ĵ ?

2. What are the components of a vector A→ = 2î + 3ĵ along the directions of î + ĵ and î − ĵ ?

To calculate î + ĵ , add unit vectors, î and ĵ , using Parallelogram Law.

Add Unit Vector

|R→| = |î + ĵ| = √1² + 1² + 2 × 1 × 1 cosπ/2 = √2

tan α = 1 sinπ/2/1 + 1 cosπ/2 = 1

α = tan^-11 = π/4

To calculate î - ĵ , add unit vectors, î and -ĵ , using Parallelogram Law.

Add Vector Law

|R→| = |î - ĵ| = √1² + 1² + 2 × 1 × 1 cos(-π/2) = √2

tan α = 1 sin(-π/2)/1 + 1 cos(-π/2) = -1

α = tan^-1-1 = -π/4

We have to calculate components of A→ = 2î + 3ĵ , along (î + ĵ) and (î − ĵ).

Method −

Extend the (î + ĵ) and (î − ĵ) vectors [obtained from the last part] along their directions to create new X and Y-axis respectively. (As shown in the figure below)

Draw perpendiculars from arrow head onto the new X and Y axes as shown in the figure. (We follow the same process of resolution as we normally do, only with one change of choosing new directions as X and Y)

Calculate the magnitude of A→ and the angle between A→ and the new X-axis.
|A→| = √2² + 3² = √13
Angle = tan^-11.5 - (π/4)

Component Magnitude

Magnitude of component along (î + ĵ) [the new + X axis] = √13 cos[tan^-11.5 - (π/4)] = 3.53

Magnitude of component along (î - ĵ) [the new - Y axis] = √13 sin[tan^-11.5 - (π/4)] = -0.707

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