Question

A vector of magnitude 3 making equal angles with the $x$ and $y$ axes and perpendicular to the $z$ axis is:

• A. $\hat{i} + 2\sqrt{2}\hat{j}$
• B. $3\hat{k}$
• C. $\frac{3}{\sqrt{2}}\hat{i} + \frac{3}{\sqrt{2}}\hat{j}$
• D. $3\hat{i} + 3\hat{j} + 3\hat{k}$

Hint:

You can quickly eliminate incorrect options by computing their magnitude! The magnitude of option (C) is $\sqrt{(\frac{3}{\sqrt{2}})^2 + (\frac{3}{\sqrt{2}})^2} = \sqrt{\frac{9}{2} + \frac{9}{2}} = \sqrt{9} = 3$. This saves time in multiple-choice exams!

Answer 1

The Correct Option is C

Concept: Let a vector be $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$.
• If a vector is perpendicular to the $z$-axis, its $z$-component must be 0 ($z=0$).
• If it makes equal angles with the $x$ and $y$ axes, its direction cosines are equal ($\cos\alpha = \cos\beta$), which means the absolute values of its $x$ and $y$ components are identical ($|x| = |y|$).

Step 1: Define the vector using the condition of perpendicularity.
Since the vector is perpendicular to the $z$-axis (which has direction vector $\hat{k}$): \[ \vec{r} \cdot \hat{k} = 0 \quad \Rightarrow \quad z = 0 \] So the vector lies purely in the $xy$-plane and can be represented as: \[ \vec{r} = x\hat{i} + y\hat{j} \]

Step 2: Apply the equal angle condition.
The vector makes equal angles with the positive directions of the $x$-axis and $y$-axis. Therefore: \[ x = y \] Substituting this back into our vector formulation: \[ \vec{r} = x\hat{i} + x\hat{j} \]

Step 3: Use the magnitude condition to solve for $x$.
We are given that the total magnitude of the vector is 3: \[ |\vec{r}| = \sqrt{x^2 + x^2} = 3 \] \[ \sqrt{2x^2} = 3 \quad \Rightarrow \quad \sqrt{2}|x| = 3 \] Assuming positive direction coordinates for components: \[ x = \frac{3}{\sqrt{2}} \] Since $y = x$, we have $y = \frac{3}{\sqrt{2}}$ as well.

Step 4: Reconstruct the vector.
Substituting the values of $x$ and $y$ back into the vector equation: \[ \vec{r} = \frac{3}{\sqrt{2}}\hat{i} + \frac{3}{\sqrt{2}}\hat{j} \] This matches option (C).