Question

The area of parallelogram formed by vectors \(\overset{⃗}{P} = 2\hat{i}-\hat{j}+5\hat{k}\) and \(\overset{⃗}{Q} = 3\hat{i}-2\hat{j}+4\hat{k}\) is

• A. \(\sqrt{72}\)
• B. \(\sqrt{86}\)
• C. \(\sqrt{104}\)
• D. \(\sqrt{240}\)

Hint:

Area equals the magnitude of the cross product P x Q.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The area of a parallelogram with adjacent sides \(\vec P\) and \(\vec Q\) equals the magnitude of their cross product, \(|\vec P \times \vec Q|\).

Step 2: Compute the cross product:
\[ \vec P \times \vec Q = \begin{vmatrix} \hat i & \hat j & \hat k \\ 2 & -1 & 5 \\ 3 & -2 & 4 \end{vmatrix} \]
\(\hat i\) component: \((-1)(4) - (5)(-2) = -4 + 10 = 6\).
\(\hat j\) component: \(-[(2)(4) - (5)(3)] = -[8 - 15] = 7\).
\(\hat k\) component: \((2)(-2) - (-1)(3) = -4 + 3 = -1\).
So \(\vec P \times \vec Q = 6\hat i + 7\hat j - \hat k\).

Step 3: Magnitude:
\[ |\vec P \times \vec Q| = \sqrt{36 + 49 + 1} = \sqrt{86} \]

Step 4: Why the other options are wrong.
\(\sqrt{72}\), \(\sqrt{104}\) and \(\sqrt{240}\) come from sign mistakes in the components, for example taking the \(\hat j\) component with the wrong sign.

Final Answer:
The area is \(\sqrt{86}\) square units, option (B). \[ \boxed{\sqrt{86}} \]