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A non-zero vector $\vec{a}$ is parallel to the line of intersection of the plane determined by vectors $\hat{i}$ and $\hat{i}+\hat{j}$ and the plane determined by vectors $\hat{i}-\hat{j}$ and $\hat{i}+\hat{k}$.Find the angle between $\vec{a}$ and vector $\hat{i}-2\hat{j}+2\hat{k}.$


I dont know how to find the equation of the plane determined by vectors $\hat{i}$ and $\hat{i}+\hat{j}$ and how to find the equation of the plane determined by vectors $\hat{i}-\hat{j}$ and $\hat{i}+\hat{k}$.

I got stuck and could not solve further.Please help me.Thanks

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The cross-product of two vectors is perpendicular to the plane containing the two vectors.

So the vectors $\vec{n_2}=(\hat i-\hat j)\times(\hat i+\hat k)$ and $\vec{n_2}=(\hat i-\hat j)\times(\hat i+\hat k)$ , must be the normal vectors to the two planes.

enter image description here

The above image is really helpful in visualising that $\vec{a}=\lambda (\vec{n_1}\times\vec{n_2})$ is along the line of intersection of the two planes.

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  • $\begingroup$ This image is very helpful sir.Thank you. $\endgroup$ – Vinod Kumar Punia Nov 6 '15 at 8:52
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Hint: The vectors $n_1=\hat i\times(\hat i+\hat j)$ and $n_2=(\hat i-\hat j)\times(\hat i+\hat k)$ are normal vectors to the planes. Then what $n_1\times n_2$ could be?

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