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Show that the points (1, 1, 1) and (-3, 0, 1) are equidistant from the plane `bar r (3bari+4barj-12bark)+13=0`

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#### Solution

Let p1 and p2 be the distances of (1, 1, 1) and (-3, 0, 1) from the plane `barr`

`(3hati+4hatj-12hatk)+13=0`

cosider `p_1=|((hati+hatj+hatk).(3hati+4hatj-12hatk)+13)/sqrt(3^2+4^2+(-12)^2)|`

`=|(1(3)+1(4)+1(-12)+13)/sqrt(9+16+144)|=|(3+4-12+13)/13=8/13.....(i)`

also `p_2=|((13hati+0hatj+hatk).(3hati+4hatj-12hatk)+13)/sqrt(3^2+4^2+(-12)^2)|`

`=|(-3(3)+0+1(-12)+13)/sqrt(9+16+144)|=|(-9-12+13)/13=8/13 ....(ii)`

From (i) and (ii),

p_{1} = p_{2}

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