For any vector a→, the value of(a→×i^)2+(a→×j^)2+(a→×k^)2=

(a→×i^)2=(a→×i^)⋅(a→×i^)=a→⋅a→a→⋅ii^⋅a→1=|a→|2−(a→⋅i^)2Σ(a→×i^)2=3|a→|2−Σ(a→⋅i^)2Since a→⋅i^=a1, etc, Ifa→=a1i^+a2j^+a3k^.

For any vector a→, the value of(a→×i^)2+(a→×j^)2+(a→×k^)2=

(a→×i^)2=(a→×i^)⋅(a→×i^)=a→⋅a→a→⋅ii^⋅a→1=|a→|2−(a→⋅i^)2Σ(a→×i^)2=3|a→|2−Σ(a→⋅i^)2Since a→⋅i^=a1, etc, Ifa→=a1i^+a2j^+a3k^.

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For any vector $\vec{a}$ , the value of
$(\vec{a} \times \hat{i})^{2} + (\vec{a} \times \hat{j})^{2} + (\vec{a} \times \hat{k})^{2} =$

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$| \vec{a} |^{2}$

$2 | \vec{a} |^{2}$

$3 | \vec{a} |^{2}$

$4 | \vec{a} |^{2}$

answer is B.

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Detailed Solution

$\begin{array}{l} (\vec{a} \times \hat{i})^{2} = (\vec{a} \times \hat{i}) \cdot (\vec{a} \times \hat{i}) = |\begin{array}{cc} \vec{a} \cdot \vec{a} & \vec{a} \cdot i \\ \hat{i} \cdot \vec{a} & 1 \end{array}| \\ = | \vec{a} |^{2} - (\vec{a} \cdot \hat{i})^{2} \\ Σ (\vec{a} \times \hat{i})^{2} = 3 | \vec{a} |^{2} - Σ (\vec{a} \cdot \hat{i})^{2} \end{array}$