Question

The angle made by the resultant vector of two vectors \( 2\hat{i} + 3\hat{j} + 4\hat{k} \) and \( 2\hat{i} - 7\hat{j} - 4\hat{k} \) with the x-axis is:

• A. \( 60^\circ \)
• B. \( 45^\circ \)
• C. \( 90^\circ \)
• D. \( 120^\circ \)

Hint:

To determine the angle a vector makes with the x-axis, use the formula \( \tan \theta = \frac{|y|}{|x|} \).

Answer 1

The Correct Option is B

Step 1: Determine the Resultant Vector Adding the given vectors: \[ \mathbf{R} = (2+2) \hat{i} + (3-7) \hat{j} + (4-4) \hat{k}. \] \[ \mathbf{R} = 4\hat{i} - 4\hat{j}. \]
Step 2: Find the Angle with the X-Axis Using the formula: \[ \tan \theta = \frac{|\text{coefficient of } j|}{|\text{coefficient of } i|}. \] \[ \tan \theta = \frac{4}{4} = 1. \] \[ \theta = 45^\circ. \] % Final Answer Thus, the correct answer is option (2): \( 45^\circ \).

Answer 2

Step 1: Calculate the Resultant Vector
To find the resultant, sum the components of the vectors:
\[ \mathbf{R} = (2 + 2) \hat{i} + (3 - 7) \hat{j} + (4 - 4) \hat{k}. \] Simplifying the terms:
\[ \mathbf{R} = 4 \hat{i} - 4 \hat{j} + 0 \hat{k} = 4\hat{i} - 4\hat{j}. \]
Step 2: Calculate the Angle Made with the X-Axis
The angle \( \theta \) between vector \( \mathbf{R} \) and the x-axis is found using:
\[ \tan \theta = \frac{|\text{component along } j|}{|\text{component along } i|} = \frac{4}{4} = 1. \] Thus:
\[ \theta = \tan^{-1} (1) = 45^\circ. \]
Final Answer:
Therefore, the vector makes an angle of:
\[ \boxed{45^\circ} \] with the x-axis.