Solveeit Logo
Login

Question

Mathematics Question on Vector Algebra

For any vector xx, where i^,j^,k^\hat{ i }, \hat{ j }, \hat{ k } have their usual meanings the value of (x×i^)2+(x×j^)2+(x×k^)2( x \times \hat{ i })^{2}+( x \times \hat{ j })^{2}+( x \times \hat{ k })^{2} where i^,j^,k\hat{ i }, \hat{ j }, \mathbf { k } have their usual meanings, is equal to

A

∣x∣2| x |^{2}

B

2∣x∣22\left|{x}\right|^{2}

C

3∣x∣23\left|{x}\right|^{2}

D

4∣x∣24\left|{x}\right|^{2}

Answer

2∣x∣22\left|{x}\right|^{2}

Explanation

Solution

Let x=αi^+βj^+γkx =\alpha \hat{ i }+\beta \hat{ j }+\gamma k
Then, x×i^=−βk^+γj^x \times \hat{ i }=-\beta \hat{ k }+\gamma \hat{ j }
x×j^=k^−γi^x \times \hat{ j }=\hat{ k }-\gamma \hat{ i }
x×k=−aj+βi^x \times k =- a j +\beta \hat{ i }
Now, (x×i^)2=(x×i^)⋅(x×i^)( x \times \hat{ i })^{2}=( x \times \hat{ i }) \cdot( x \times \hat{ i })
=(−βk^+γj^)⋅(−βk^+γj^)=(-\beta \hat{k}+\gamma \hat{j}) \cdot(-\beta \hat{k}+\gamma \hat{j})
=β2+γ2=\beta^{2}+\gamma^{2}
Similarly, (x×j^)2=α2+γ2( x \times \hat{j})^{2}=\alpha^{2}+\gamma^{2}
and (x×K^)2=α2+β2(x \times \hat{K} )^{2}=\alpha^{2}+\beta^{2}
∴(x×i^)2+(x×j^)2+(x×K^)2\therefore(x \times \hat{i})^{2}+(x \times \hat{j})^{2}+(x \times \hat{K} )^{2}
=β2+γ2+α2+γ2+α2+β2=\beta^{2}+\gamma^{2}+\alpha^{2}+\gamma^{2}+\alpha^{2}+\beta^{2}
=2(α2+β2+γ2)=2∣x∣2=2\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)=2|x|^{2}