JEE PYQ: Vector Algebra - Question ID 42a3e5fdc289 (JEE Main 2023)

In vector algebra, any vector $\vec{A}$ lying in the $x$-$y$ plane can be resolved into its Cartesian components along the $x$ and $y$ axes. If the vector makes an angle $\theta$ with the $y$-axis, then:

• The $y$-component is given by $A_y = |\vec{A}| \cos \theta$.
• The $x$-component is given by $A_x = |\vec{A}| \sin \theta$.

These relations arise from the definitions of cosine and sine in a right triangle formed by the vector and its projections on the axes.

Given:

• Angle with $y$-axis, $\theta = 30^\circ$.
• Magnitude of $y$-component, $A_y = 2\sqrt{3}$.

Step 1: Find the magnitude of the vector $|\vec{A}|$.

Using the relation for the $y$-component: $A_y = |\vec{A}| \cos \theta$ Substitute the known values: $2\sqrt{3} = |\vec{A}| \cos 30^\circ$ We know that $\cos 30^\circ = \frac{\sqrt{3}}{2}$, so: $2\sqrt{3} = |\vec{A}| \cdot \frac{\sqrt{3}}{2}$ Solve for $|\vec{A}|$: $|\vec{A}| = \frac{2\sqrt{3}}{\frac{\sqrt{3}}{2}} = 2\sqrt{3} \cdot \frac{2}{\sqrt{3}} = 4$

Step 2: Find the magnitude of the $x$-component $A_x$.

Using the relation for the $x$-component: $A_x = |\vec{A}| \sin \theta$ Substitute the known values: $A_x = 4 \cdot \sin 30^\circ$ We know that $\sin 30^\circ = \frac{1}{2}$, so: $A_x = 4 \cdot \frac{1}{2} = 2$

Conclusion:

The magnitude of the $x$-component of the vector is $2$, which corresponds to option B.

Students often confuse the angle given with respect to the $y$-axis as the angle with the $x$-axis. This leads to incorrect use of sine and cosine functions. Always:

• Draw a clear diagram to visualize the vector and its components.
• Identify which axis the angle is measured from (here, it is the $y$-axis).
• Use $A_y = |\vec{A}| \cos \theta$ and $A_x = |\vec{A}| \sin \theta$ when the angle is with the $y$-axis.
• Double-check the trigonometric values for standard angles (e.g., $30^\circ$, $45^\circ$, $60^\circ$).

A common mistake is to assume the angle is with the $x$-axis and use $A_x = |\vec{A}| \cos \theta$, which would yield an incorrect result.