Question

The vector \(\overset{̄}{r}\) whose magnitude is \(3\sqrt{2}\) units and which makes angles of \(\frac{π}{4}\) and \(\frac{π}{2}\) with the positive y- and z-axes respectively is....

• A. \(\hat{i}\pm 3\hat{j}\)
• B. \(\hat{i}\pm \hat{j}\)
• C. \(-\hat{i}\pm \hat{j}\)
• D. \(\pm 3\hat{i}+3\hat{j}\)

Hint:

Use cos^2 sum = 1 to find the x direction cosine.

Answer 1

The Correct Option is D

Step 1: Use the magnitude check:
The vector lies in the $xy$-plane because the angle with $z$ is $90^\circ$. So $\vec r=x\hat i+y\hat j$.

Step 2: Use the y-axis angle:
$y=r\cos45^\circ=3\sqrt2\cdot\frac{1}{\sqrt2}=3$.

Step 3: Use the magnitude:
$x^2+y^2=18$ gives $x^2=9$, so $x=\pm3$.

Step 4: Result:
$\pm3\hat i+3\hat j$. Option (D).

Final Answer:
The vector lies in the xy-plane, with y = 3 and x = plus or minus 3. \[ \boxed{D} \]