JEE PYQ: Vector Algebra - Question ID 20cb7b2289b3 (JEE Main 2023)

In vector algebra, the cross product of two vectors $\overrightarrow{P}$ and $\overrightarrow{Q}$ produces a third vector $\overrightarrow{P} \times \overrightarrow{Q}$ that is perpendicular to both $\overrightarrow{P}$ and $\overrightarrow{Q}$. Its magnitude is $|\overrightarrow{P} \times \overrightarrow{Q}| = |\overrightarrow{P}|\,|\overrightarrow{Q}|\,\sin\theta,$ where $\theta$ is the angle between $\overrightarrow{P}$ and $\overrightarrow{Q}$. A unit vector in the direction of $\overrightarrow{P} \times \overrightarrow{Q}$ is obtained by dividing the cross product by its own magnitude: $\hat{n} = \frac{\overrightarrow{P} \times \overrightarrow{Q}}{|\overrightarrow{P} \times \overrightarrow{Q}|}.$

1. Compute the cross product $\overrightarrow{P} \times \overrightarrow{Q}$.
Given $\overrightarrow{P} = 3\widehat{i} + \sqrt{3}\,\widehat{j} + 2\widehat{k}, \quad \overrightarrow{Q} = 4\widehat{i} + \sqrt{3}\,\widehat{j} + 2.5\widehat{k},$ we use the determinant formula for the cross product: $$\overrightarrow{P} \times \overrightarrow{Q} = \begin{vmatrix} \widehat{i} & \widehat{j} & \widehat{k} \\ 3 & \sqrt{3} & 2 \\ 4 & \sqrt{3} & 2.5 \\ \end{vmatrix}.$$ Expanding along the first row: $= \widehat{i}\bigl(\sqrt{3}\cdot2.5 - 2\cdot\sqrt{3}\bigr) - \widehat{j}\bigl(3\cdot2.5 - 2\cdot4\bigr) + \widehat{k}\bigl(3\cdot\sqrt{3} - \sqrt{3}\cdot4\bigr).$ Simplify each component: $= \widehat{i}\bigl(2.5\sqrt{3} - 2\sqrt{3}\bigr) - \widehat{j}\bigl(7.5 - 8\bigr) + \widehat{k}\bigl(3\sqrt{3} - 4\sqrt{3}\bigr) = \sqrt{3}\,\widehat{i} + 0.5\,\widehat{j} - \sqrt{3}\,\widehat{k}.$

2. Compare with the given form.
The problem states that the unit vector is $\frac{1}{x}\bigl(\sqrt{3}\,\widehat{i} + \widehat{j} - 2\sqrt{3}\,\widehat{k}\bigr).$ Our computed cross product is $\sqrt{3}\,\widehat{i} + 0.5\,\widehat{j} - \sqrt{3}\,\widehat{k}.$ Notice that the given form is exactly twice our result: $\sqrt{3}\,\widehat{i} + \widehat{j} - 2\sqrt{3}\,\widehat{k} = 2\bigl(\sqrt{3}\,\widehat{i} + 0.5\,\widehat{j} - \sqrt{3}\,\widehat{k}\bigr).$ Hence the factor $x$ must account for both the scaling to a unit vector and this factor of $2$.

3. Compute the magnitude of $\overrightarrow{P} \times \overrightarrow{Q}$.
$|\overrightarrow{P} \times \overrightarrow{Q}| = \sqrt{(\sqrt{3})^2 + (0.5)^2 + (-\sqrt{3})^2} = \sqrt{3 + 0.25 + 3} = \sqrt{6.25} = 2.5.$

4. Form the unit vector.
$$\hat{n} = \frac{\overrightarrow{P} \times \overrightarrow{Q}}{|\overrightarrow{P} \times \overrightarrow{Q}|} = \frac{\sqrt{3}\,\widehat{i} + 0.5\,\widehat{j} - \sqrt{3}\,\widehat{k}}{2.5} = \frac{2}{5}\bigl(\sqrt{3}\,\widehat{i} + 0.5\,\widehat{j} - \sqrt{3}\,\widehat{k}\bigr).$$ Multiply numerator and denominator by $2$ to match the given form: $= \frac{1}{\tfrac{5}{2}}\bigl(\sqrt{3}\,\widehat{i} + \widehat{j} - 2\sqrt{3}\,\widehat{k}\bigr) = \frac{1}{2.5}\bigl(\sqrt{3}\,\widehat{i} + \widehat{j} - 2\sqrt{3}\,\widehat{k}\bigr).$ But $2.5 = \tfrac{5}{2}$, so we can also write $\hat{n} = \frac{1}{\tfrac{5}{2}}\bigl(\sqrt{3}\,\widehat{i} + \widehat{j} - 2\sqrt{3}\,\widehat{k}\bigr) = \frac{2}{5}\bigl(\sqrt{3}\,\widehat{i} + \widehat{j} - 2\sqrt{3}\,\widehat{k}\bigr).$ Comparing with the problem’s expression $\frac{1}{x}\bigl(\sqrt{3}\,\widehat{i} + \widehat{j} - 2\sqrt{3}\,\widehat{k}\bigr),$ we see $x = \tfrac{5}{2}\times 2 = 5$ is not consistent. Instead, observe that the given form is already scaled so that its numerator is twice the cross product. Hence the magnitude of the numerator is $|\sqrt{3}\,\widehat{i} + \widehat{j} - 2\sqrt{3}\,\widehat{k}| = \sqrt{3 + 1 + 12} = \sqrt{16} = 4.$ Therefore the unit vector is $\frac{1}{4}\bigl(\sqrt{3}\,\widehat{i} + \widehat{j} - 2\sqrt{3}\,\widehat{k}\bigr),$ which matches the problem’s $\tfrac{1}{x}(\cdots)$. Thus $x = 4$.

1. Sign errors in the cross product: Students often mix up the signs when expanding the determinant. Always double-check the cyclic order $\widehat{i}\to\widehat{j}\to\widehat{k}$. 2. Magnitude miscalculation: Forgetting to square each component or miscounting the squares leads to wrong magnitudes. Verify each term carefully. 3. Scaling confusion: The problem gives a vector that is a scaled version of the cross product. One must relate the given form back to the actual cross product before computing the unit vector.