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What will be the unit vector in XY-plane, which makes an angle of 30° with the positive direction of x-axis?

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AAI ATC Junior Executive 25 March 2021 Official Paper (Shift 2)
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  1. \(\frac{1}{\sqrt{3}} \hat{\imath}+\frac{1}{\sqrt{2}} \hat{\jmath}\)
  2. \(\frac{\sqrt{3}}{2} \hat{\imath}+\frac{1}{2} \hat{\jmath}\)
  3. \(\frac{1}{\sqrt{3}} \hat{\imath}+\frac{1}{2} \hat{\jmath}\)
  4. \(\frac{1}{\sqrt{2}} \hat{\jmath}\)

Answer (Detailed Solution Below)

Option 2 : \(\frac{\sqrt{3}}{2} \hat{\imath}+\frac{1}{2} \hat{\jmath}\)
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Detailed Solution

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Concept:

A unit vector that makes an angle α with the positive direction of the x-axis and an angle β with the positive direction of the y-axis is given by cos(α)î + cos(β)ĵ 

Explanation:

Given:

The given unit vector makes an angle of 30° with the positive direction of the x-axis.

Angle with the positive direction of x-axis = 30°

⇒ Angle with the positive direction of the y-axis = 60°

∴ The required unit vector is cos(30°)î + cos(60°)ĵ 

= \(\frac{\sqrt{3}}{2} \hat{\imath}+\frac{1}{2} \hat{\jmath}\)

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