यदि \(\vec{a}\ =\ \hat{i}\ +\ 2\hat{j}\ +\ 2\hat{k}\) और \(\vec{b}\ =\ 3\hat{i}\ +\ 5\hat{j}\ +\ \sqrt{2}\hat{k}\) हो तो a की दिशा में एक सदिश और |b| के रूप में परिमाण क्या है?

  1. \(\frac{1}{2}(3 \hat{i}\ +\ 5\hat{j}\ +\ \hat{k})\)
  2. \(2( \hat{i}\ -\ 2\hat{j}\ +\ 2\hat{k})\)
  3. \(\frac{1}{2}(3 \hat{i}\ -\ 5\hat{j}\ +\ \hat{k})\)
  4. \(2( \hat{i}\ +\ 2\hat{j}\ +\ 2\hat{k})\)

Answer (Detailed Solution Below)

Option 4 : \(2( \hat{i}\ +\ 2\hat{j}\ +\ 2\hat{k})\)
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Detailed Solution

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संकल्पना:

 \(\vec{q}\) की दिशा में परिमाण |p| का सदिश \(\vec{r}\) निम्न द्वारा दिया गया है

\(\vec{r}\ =\ |p|.\hat{q}\ =\ |p|.\frac{\hat{q}}{|q|}\)

यदि \(\vec{a} = {a_1}\hat{i} \ + \ {a_2}\hat{j} \ + \ {a_3}\hat{k}\) हो तो \(\vec{a}\) के परिमाण को निम्न रूप में लिखा जाता है

\(|\vec{a}|\ =\ \sqrt{a_1^2\ +\ a_2^2\ +\ a_3^2}\)

गणना:

माना आवश्यक सदिश \(\vec{c}\)

\(\vec{a}\ =\ \hat{i}\ +\ 2\hat{j}\ +\ 2\hat{k}\)

\(\vec{b}\ =\ 3\hat{i}\ +\ 5\hat{j}\ +\ \sqrt{2}\hat{k}\)

दिशा a में एक सदिश और परिमाण |b| निम्न द्वारा दिया गया है

\(\vec{c}\ =\ |b|.\hat{a}\ =\ |b|.\frac{\hat{a}}{|a|}\)

\(\vec{c}\ =\ \sqrt{3^2\ +\ 5^2\ +\ (\sqrt{2})^2}.\frac{\hat{\hat{i}\ +\ 2\hat{j}\ +\ 2\hat{k}}}{\sqrt{1^2\ +\ 2^2\ +\ 2^2}}\)

\(\vec{c}\ =\ 6.\frac{\hat{\hat{i}\ +\ 2\hat{j}\ +\ 2\hat{k}}}{3}\)

\(\vec{c}\ =\ 2({\hat{\hat{i}\ +\ 2\hat{j}\ +\ 2\hat{k}}})\)

अत: विकल्प 4 सही है।

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