Q.Write down a unit vector in XY-plane, making an angle of $30^\circ$ with the positive direction of x-axis.

The key idea is that any vector in the XY-plane can be written using its direction cosines. A unit vector making an angle $\theta$ with the positive x‑axis has components $(\cos\theta, \sin\theta)$.

Here $\theta = 30^\circ$, so:

• $\cos 30^\circ = \frac{\sqrt{3}}{2}$
• $\sin 30^\circ = \frac{1}{2}$

Thus the required unit vector is:

Why Direction Vectors Work

Any vector in the XY-plane can be thought of as an arrow from the origin to a point $(x, y)$. For a unit vector (length = 1), the tip of the arrow lies exactly on the unit circle. The angle $\theta$ measured counterclockwise from the positive x-axis gives the coordinates of that point as $(\cos\theta, \sin\theta)$. So the vector itself is $\cos\theta\,\hat{i} + \sin\theta\,\hat{j}$.

This is the cleanest way to write a direction vector — no square roots, no Pythagoras needed after you know the angle.

Step-by-step

1. Identify the angle. The problem says $30^\circ$ with the positive x-axis. In standard position, that’s $\theta = 30^\circ$.

2. Write the components using cosine and sine.

   For any angle $\theta$, a unit vector in the XY-plane is

3. Plug in $\theta = 30^\circ$.

4. Assemble the vector.

5. Verify the length.

So it is indeed a unit vector.

Method: Writing a Unit Vector at a Given Angle in the XY-Plane

Use this whenever you need a unit vector in the plane making a stated angle with the positive $x$-axis.

Steps

Step 1: Recall the standard form

A unit vector in the XY-plane at angle $\theta$ (measured anticlockwise from the positive $x$-axis) is

This is the point where the terminal ray meets the unit circle, so its length is automatically $1$.

Step 2: Substitute the given angle

Replace $\theta$ with the stated angle and use exact trig values (e.g. $\cos30^\circ=\tfrac{\sqrt3}{2}$, $\sin30^\circ=\tfrac12$). Cosine pairs with $\hat i$, sine with $\hat j$.

Step 3: (Optional) confirm it is a unit vector

Check $\sqrt{\cos^2\theta+\sin^2\theta}=1$ — guaranteed by the Pythagorean identity, so it is just a quick sanity check.

Common Mistakes

Mistake 1: Swapping sine and cosine

Why it's wrong: writing $\tfrac12\hat i+\tfrac{\sqrt3}{2}\hat j$ is the unit vector at $60^\circ$, not $30^\circ$. Correct approach: with the angle measured from the positive $x$-axis, cosine goes with $\hat i$ and sine with $\hat j$.

Mistake 2: Forgetting to make it a unit vector

Why it's wrong: the question demands length $1$. Correct approach: $\cos\theta\,\hat i+\sin\theta\,\hat j$ already has magnitude $1$ — no extra scaling.

Mistake 3: Adding a $\hat k$ component

Why it's wrong: the vector must lie in the XY-plane, so its $z$-component is $0$. Correct approach: keep it two-dimensional.