If the centroid of a tetrahedron OABC is (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.

Engineering

Mathematics

Vector Addition and Subtraction

Section Formulae and Centres of a Triangle

Distance from a Plane

Question

If the centroid of a tetrahedron OABC is (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 OABC.

Let

\vec{a}, \vec{b}, \vec{c}, \vec{g}

be the position vectors of the points A,B,C,G respectively w.r.t. O.

then

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

\vec{b} = \hat{i} + b \hat{j} + 2 \hat{k}

\vec{c} = 2 \hat{i} + \hat{j} + c \hat{k}

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

By formula of centroid of a tetrahedron,

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

∴ 4 \vec{g} = \vec{a} + \vec{b} + \vec{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}

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