Question

Let \(\overset{̄}{a} = \hat{i}+\hat{j}+\hat{k}\), \(\overset{̄}{b} = \hat{i}-3\hat{j}+2\hat{k}\) and \(\overset{̄}{c} = 3\hat{i}-2\hat{k}\). If a vector \(\overset{̄}{p}\) satisfies the conditions \(\overset{̄}{p}\cdot \overset{̄}{c} = 0\) and \(\overset{̄}{p}\times \overset{̄}{a} = \overset{̄}{b}\times \overset{̄}{a}\), then the value of \(|\overset{̄}{p}| =\)...

• A. \(\sqrt{13}\)
• B. \(\sqrt{14}\)
• C. \(\sqrt{17}\)
• D. \(\sqrt{19}\)

Hint:

p x a = b x a means (p - b) is parallel to a, so p = b + lambda a.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The condition \(\bar p \times \bar a = \bar b \times \bar a\) gives \((\bar p - \bar b)\times\bar a = 0\). Two vectors with zero cross product are parallel, so \(\bar p - \bar b = \lambda\bar a\).

Step 2: Write p:
\(\bar p = \bar b + \lambda\bar a = (1+\lambda)\hat i + (-3+\lambda)\hat j + (2+\lambda)\hat k\).

Step 3: Use the dot condition:
\(\bar p\cdot\bar c = 3(1+\lambda) + 0 - 2(2+\lambda) = 3 + 3\lambda - 4 - 2\lambda = \lambda - 1 = 0\), so \(\lambda = 1\).

Step 4: Magnitude:
\(\bar p = 2\hat i - 2\hat j + 3\hat k\), so \(|\bar p| = \sqrt{4 + 4 + 9} = \sqrt{17}\).

Step 5: Why the other options are wrong.
\(\sqrt{13}\), \(\sqrt{14}\) and \(\sqrt{19}\) arise from arithmetic slips in the components, such as using \(\lambda = 0\) or \(\lambda = 2\). With \(\lambda = 0\), \(\bar p = \bar b\) has magnitude \(\sqrt{14}\), but then \(\bar p\cdot\bar c = 3 - 4 \ne 0\).

Final Answer:
\(|\bar p| = \sqrt{17}\), option (C). \[ \boxed{\sqrt{17}} \]