If the centroid of a tetrahedron O A B C is \left(\right. 1 , 2 , - 1 \left.\right) where A \left(\right. a , 2 , 3 \left.\right) , B \left(\right. 1 , b , 2 \left.\right) , C \left(\right. 2 , 1 , c \left.\right), find the distance of P \left(\right. a , b , c \left.\right) from origin.

Engineering

Mathematics

Line of Greatest Slope

Question

If the centroid of a tetrahedron

O A B C

(1, 2, - 1)

where

A (a, 2, 3), B (1, b, 2), C (2, 1, c)

, find the distance of

P (a, b, c)

from origin.

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Let

G = (1, 2, - 1)

be the centroid of the tetrahedron

O A B C

Let

\overset{ˉ}{a}, \overset{ˉ}{b}, \overset{ˉ}{c}, \overset{ˉ}{g}

be the position vectors of the points

A, B, C, G

respectively w.r.t.

O

then

\overset{ˉ}{a} = a \hat{i} + 2 \hat{j} + 3 \hat{k}

\overset{ˉ}{b} = \hat{i} + b \hat{j} + 2 \hat{k}

\overset{ˉ}{c} = 2 \hat{i} + \hat{j} + c \hat{k}

\overset{ˉ}{g} = \hat{i} + 2 \hat{j} - \hat{k}

By formula of centroid of a tetrahedron,

\overset{ˉ}{g} = \frac{\overset{ˉ}{0} + \overset{ˉ}{a} + \overset{ˉ}{b} + \overset{ˉ}{c}}{4}

∴ 4 \overset{ˉ}{g} = \overset{ˉ}{a} + \overset{ˉ}{b} + \overset{ˉ}{c}

∴ 4 (\hat{i} + 2 \hat{j} - \hat{k}) = (a \hat{i} + 2 \hat{j} + 3 \hat{k}) + (\hat{i} + b \hat{j} + 2 \hat{k}) + (2 \hat{i} + \hat{j} + c \hat{k})

∴ 4 \hat{i} + 8 \hat{j} - 4 \hat{k} = (a + 1 + 2) \hat{i} + (2 + b + 1) \hat{j} + (3 + 2 + c) \hat{k}

∴ 4 \hat{i} + 8 \hat{j} - 4 \hat{k} = (a + 3) \hat{i} + (b + 3) \hat{j} + (c + 5) \hat{k}

By equality of vectors

a + 3 = 4, b + 3 = 8, c + 5 = - 4

∴ a = 1, b = 5, c = - 9

∴ P (a, b, c) = (1, 5, - 9)

Distance of

P

from origin

= \sqrt{1^{2} + 5^{2} + (- 9)^{2}}

= \sqrt{1 + 25 + 81}

= \sqrt{107}

units

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