Question

If a is the angle made by the vector \(\overrightarrow{a}=5\hat{i}+3\hat{j}+4\hat{k}\) with the positive x-axis, then cosa =

• A. \(\frac{5}{12}\)
• B. \(\frac{1}{2}\)
• C. \(\frac{\sqrt2}{2}\)
• D. \(\frac{\sqrt5}{5}\)
• E. \(\frac{\sqrt2}{10}\)

Answer 1

The Correct Option is C

Step 1: Use the direction cosine formula  
The cosine of the angle \( a \) made by a vector \( \overrightarrow{a} \) with the positive x-axis is given by: \[ \cos a = \frac{a_x}{|\overrightarrow{a}|} \] where \( a_x \) is the x-component of \( \overrightarrow{a} \).

Step 2: Compute the magnitude of \( \overrightarrow{a} \) 
Given: \[ \overrightarrow{a} = 5\hat{i} + 3\hat{j} + 4\hat{k} \] The magnitude is: \[ |\overrightarrow{a}| = \sqrt{5^2 + 3^2 + 4^2} \] \[ = \sqrt{25 + 9 + 16} = \sqrt{50} = 5\sqrt{2} \]

Step 3: Compute \( \cos a \) 
\[ \cos a = \frac{5}{5\sqrt{2}} \] \[ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]

Final Answer: \( \cos a \) is \( \frac{\sqrt{2}}{2} \).