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प्रश्न
Find the Cartesian equation of the following planes:
`vecr.(hati + hatj-hatk) = 2`
उत्तर
It is given that equation of the plane is
`vecr.(hati + hatj-hatk) = 2`
For any arbitrary point P (x, y, z) on the plane, position vector `vecr`is given by,
`vecr = xhati +yhatj-zhatk`
Substituting the value of `vecr` in equation (1), we obtain
`(xhati + yhatj - zhatk).(hati + hatj - hatk) = 2`
=> x + y - z = 2
This is the Cartesian equation of the plane.
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संबंधित प्रश्न
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Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes \[\vec{r} \cdot \left( \hat{i} - \hat{j} + 2 \hat{k} \right) = 5 \text{ and } \vec{r} \cdot \left( 3 \hat{i} + \hat{j} + \hat{k} \right) = 6 .\]
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Find the vector and Cartesian equations of the plane passing through the points (2, 2 –1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above.
Find the vector equation of the plane that contains the lines `vecr = (hat"i" + hat"j") + λ (hat"i" + 2hat"j" - hat"k")` and the point (–1, 3, –4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane thus obtained.
The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is ______.
The vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.
The Cartesian equation of the plane `vec"r" * (hat"i" + hat"j" - hat"k")` = 2 is ______.
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