Determine the value of p for which  \(p(\hat{i}+\hat{j}+\hat{k})\) is a unit vector?

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  1. \(\pm\frac{1}{{\sqrt 3 }}\)
  2. \(\pm\frac{2}{{\sqrt 3 }}\)
  3. \(\pm\frac{1}{{ 3 }}\)
  4. \({{\sqrt 3 }}\)

Answer (Detailed Solution Below)

Option 1 : \(\pm\frac{1}{{\sqrt 3 }}\)
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Detailed Solution

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Given:

 \(p(\hat{i}+\hat{j}+\hat{k})\) is a unit vector.

Concept:

A unit vector of a vector is a vector of unit magnitude and direction as that of the original vector.

Formula:

The magnitude of a vector aî + bĵ + ck̂ is given by √(a2 + b2 +c2)

Calculation:

The magnitude of given vector  \(p(\hat{i}+\hat{j}+\hat{k})\) can be calculated as;

\(p(\hat{i}+\hat{j}+\hat{k})\)|= |p|√(12 + 12 + 12)

= |p|√3

∵ it is a unit vector:

⇒ |p|√3 = 1

⇒ |p| = 1/√3

or p = ±1/√3

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