Find the vector equation of the plane passing through the intersection of the planes $r . (i + j + k) = 6$ and $r . (2 i + 3 j + 4 k) = - 5$ and the point $(1, 1, 1)$ .

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Solution

$P l a n e 1 : \to r (^i +^j +^k) = 6 P l a n e 2 : \to r (^2 i + 3^j + 4^k) = - 5 [\to r (^i +^j +^k) - 6] + λ [\to r (^2 i + 3^j + 4^k) + 5] = 0 [(^i +^j +^k) . (^i +^j +^k) - 6] + λ [(^i +^j +^k) . (^2 i + 3^j + 4^k) + 5] = 0 - 3 + λ (14) = 0 \Rightarrow λ = \frac{3}{14} \to r [(^i +^j +^k) + λ (^2 i + 3^j + 4^k)] = 6 - 5 λ \to r (\frac{10}{7}^i + \frac{23}{16}^j + \frac{13}{7}^k) = \frac{69}{14} \to r (20^i + 23^j + 26^k) = 69 ⟶ r e q u i r e d p l a n e$

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Similar questions

r . (i + j + k) = 6

r . (2 i + 3 j + 4 k) = - 5

(1, 1, 1)

\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 and \vec{r} \cdot (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) = - 5

r . (2 ¯ i + 3 ¯ j + 4 ¯ ¯ ¯ k) = 1

r . (¯ i - ¯ j) + 4 = 0

r . (2 ¯ i - ¯ j + ¯ ¯ ¯ k) = 0

2 \hat{i} + \hat{j} - \hat{k}

\vec{r} \cdot (\hat{i} + 3 \hat{j} - \hat{k}) = 0 and \vec{r} \cdot (\hat{j} + 2 \hat{k}) = 0 .

2 ˆ i + 3 ˆ j - 7 ˆ k

r = (ˆ i - 2 ˆ j - ˆ k) + λ (3 ˆ i - 8 ˆ j + 2 ˆ k)

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