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The foot of the perpendicular drawn from the origin to the plane x + y + z = 3 is

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  1. (0, 1, 2)
  2. (0, 0, 3)
  3. (1, 1, 1)
  4. (-1, 1, 3)

Answer (Detailed Solution Below)

Option 3 : (1, 1, 1)
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Detailed Solution

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CONCEPT:

Perpendicular distance of a plane ax + by + cz + d = 0 from a point P (x1, y1, z1) is given by: \(d = \left| {\frac{{a{x_1} + b{y_1} + c{z_1} + d}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}} \right|\)

CALCULATION:

Let A(x, y, z) be the foot of the perpendicular drawn from the origin to the plane x + y + z = 3.

As we know that, the perpendicular distance of a plane ax + by + cz + d = 0 from a point P (x1, y1, z1) is given by: \(d = \left| {\frac{{a{x_1} + b{y_1} + c{z_1} + d}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}} \right|\)

So, the distance between the origin and the plane x + y + z - 3 = 0 is given by: \(d = \left| {\frac{{0 + 0 + 0 - 3}}{{\sqrt {{1^2} + {1^2} + {1^2}} }}} \right| = \frac{3}{\sqrt 3} = \sqrt 3\)
So, this means that the length of the line joining the points origin and A is √3
As we can see that from the given options, if A = (1, 1, 1) then the distance between the points origin and A is √3
Hence, correct option is 3. 
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