Find the Cartesian and vector equations of the planes through the line of intersection of the planes r⋅(i^−j^)+6=0 and r⋅(3i^+3j^−4k^)=0, which are at a unit distance from the origin.

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Find the Cartesian and vector equations of the planes through

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Find the Cartesian and vector equations of the planes through the line of intersection of the planes $r \cdot (\hat{i} - \hat{j}) + 6 = 0$ and $r \cdot (3 \hat{i} + 3 \hat{j} - 4 \hat{k}) = 0$ , which are at a unit distance from the origin.

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Michael Discussed

Find the Cartesian and vector equations of the planes through the line of intersection of the planes

r \cdot (\hat{i} - \hat{j}) + 6 = 0

and

r \cdot (3 \hat{i} + 3 \hat{j} - 4 \hat{k}) = 0

, which are at a unit distance from the origin.

7 mins ago

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7 mins ago

The equations of the given planes are

(x \hat{i} + y \hat{j} + z \hat{k}) \cdot (\hat{i} - \hat{j}) + 6 = 0

and

(x \hat{i} + y \hat{j} + z \hat{k}) \cdot (3 \hat{i} + 3 \hat{j} - 4 \hat{k}) = 0

\Rightarrow x - y + 6 = 0

and

3 x + 3 y - 4 z = 0

Any plane through their intersection is

(x - y + 6) + λ (3 x + 3 y - 4 z) = 0

\Rightarrow (1 + 3 λ) x + (3 λ - 1) y - 4 λ z + 6 = 0

..........(i)

which is at a unit distance from the origin. .... (given)

∴ \frac{6}{( 1 + 3 λ ) ^{2} + ( 3 λ - 1 ) ^{2} + ( - 4 λ ) ^{2}} = 1

\Rightarrow 34 λ^{2} + 2 = 36

\Rightarrow λ^{2} = 1

\Rightarrow λ = \pm 1

So, the required planes are

2 x + y - 2 z + 3 = 0

and

x + 2 y - 2 z - 3 = 0

In vector form, they are

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

and

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

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Question Text	Find the Cartesian and vector equations of the planes through the line of intersection of the planes $r \cdot (\hat{i} - \hat{j}) + 6 = 0$ and $r \cdot (3 \hat{i} + 3 \hat{j} - 4 \hat{k}) = 0$ , which are at a unit distance from the origin.
Answer Type	Text solution:1
Upvotes	150