Question 50 of 68beginner🔧 ApplyShort Answer2 marks

Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

Correct Answer

The required unit vector is 32i^+12j^\frac{\sqrt{3}}{2}\hat{i} + \frac{1}{2}\hat{j}.

Exercise: MISCELLANEOUS EXERCISE ON CHAPTER 10 | Q: 1 | (Page 35)
For More Understanding

Explanation

From Example 26 in the textbook, any unit vector in the XY-plane making an angle θ with the positive direction of x-axis can be expressed as r⃗=cos⁡θi^+sin⁡θj^\vec{r} = \cos\theta\hat{i} + \sin\theta\hat{j}. This is because for a unit vector, the components x and y satisfy x=cos⁡θx = \cos\theta and y=sin⁡θy = \sin\theta (since ∣r⃗∣=1|\vec{r}| = 1). Substituting θ = 30° gives the required vector.

Solution Steps

  1. Step 1: From the formula for unit vector in XY-plane: r⃗=cos⁡θi^+sin⁡θj^\vec{r} = \cos\theta\hat{i} + \sin\theta\hat{j}

  2. Step 2: Given θ = 30°, substitute the values: cos⁡30°=32\cos 30° = \frac{\sqrt{3}}{2} and sin⁡30°=12\sin 30° = \frac{1}{2}

  3. Step 3: Therefore, the required unit vector is r⃗=32i^+12j^\vec{r} = \frac{\sqrt{3}}{2}\hat{i} + \frac{1}{2}\hat{j}

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