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Solve the following.
Find the equation of the plane through the intersection of the planes and and passing through the…
Find the equation of the plane through the intersection of the planes $\vec{r} \cdot (\hat{\imath} + 3\hat{\jmath} - \hat{k}) = 9$ and $\vec{r} \cdot (2\hat{\imath} - \hat{\jmath} + \hat{k}) = 3$ and passing through the origin.
Answer
Answer
AIPlane through the intersection: $\vec r\cdot\left[(\hat\imath + 3\hat\jmath - \hat k) + \lambda(2\hat\imath - \hat\jmath + \hat k)\right] = 9 + 3\lambda$.
It passes through the origin, so $0 = 9 + 3\lambda$, giving $\lambda = -3$.
Normal: $(1-6)\hat\imath + (3+3)\hat\jmath + (-1-3)\hat k = -5\hat\imath + 6\hat\jmath - 4\hat k$.
Hence the plane is $\vec r\cdot(5\hat\imath - 6\hat\jmath + 4\hat k) = 0$.
Final answer: $\vec{r}\cdot(5\hat{\imath} - 6\hat{\jmath} + 4\hat{k}) = 0$
From ISC 2018 Mathematics Paper 1, question 15(c).
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