For any vector a , then the value of ( a × i ) 2 + ( a × j ) 2 + ( a × k ) 2 is:

Question

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

Smart Achievers · Accepted Answer

(4)

Ans. $2 \vec{a^{2}}$

Explanation:

Let

$\vec{a} = x \hat{i} + y \hat{j} + z \hat{k}$

$∴$ $\vec{a^{2}} = x^{2} + y^{2} + z^{2}$

$∴$ $\vec{a} \times \hat{i} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ x & y & z \\ 1 & 0 & 0 \end{matrix} |$

$= \hat{i} [0] - \hat{j} [- z] + \hat{k} [- y]$

$= z \hat{j} - y \hat{k}$

$∴$ ${(\vec{a} \times \hat{i})}^{2} = (z \hat{j} - y \hat{k}) (z \hat{j} - y \hat{k})$

$= y^{2} + z^{2}$

Similarly, ${(\vec{a} \times \hat{j})}^{2} = x^{2} + z^{2}$

and ${(\vec{a} \times \hat{k})}^{2} = x^{2} + y^{2}$

$∴$ ${(\vec{a} \times \hat{i})}^{2} + {(\vec{a} \times \hat{j})}^{2} + {(\vec{a} \times \hat{k})}^{2}$ $= y^{2} + z^{2} + x^{2} + z^{2} + x^{2} + y^{2}$

$= 2 (x^{2} + y^{2} + z^{2})$

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

Solution

Q.
For any vector $\vec{a}$ , then the value of ${(\vec{a} \times \hat{i})}^{2} + {(\vec{a} \times \hat{j})}^{2} + {(\vec{a} \times \hat{k})}^{2}$ is:

1 $\vec{a^{2}}$

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

3 $4 \vec{a^{2}}$

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

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Solution

Q. For any vector a→, then the value of (a→×i^)2+(a→×j^)2+(a→×k^)2 is:

1 a2→

2 3a2→

3 4a2→