If xvec{a} + yvec{b} + zvec{c} = vec{0},then | vedclass

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(A) For three points $A, B, C$ with position vectors $\vec{a}, \vec{b}, \vec{c}$ to be collinear,there must exist scalars $x, y, z$ (not all zero) suc

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$x + y + z = 0$

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(A) For three points $A, B, C$ with position vectors $\vec{a}, \vec{b}, \vec{c}$ to be collinear,there must exist scalars $x, y, z$ (not all zero) such that $x\vec{a} + y\vec{b} + z\vec{c} = \vec{0}$ and $x + y + z = 0$.This is because the condition for collinearity of three points is that the vector $\vec{AB}$ is a scalar multiple of $\vec{AC}$,i.e.,$(\vec{b} - \vec{a}) = k(\vec{c} - \vec{a})$.Rearranging this gives $\vec{b} - \vec{a} = k\vec{c} - k\vec{a}$,which implies $(k-1)\vec{a} + \vec{b} - k\vec{c} = \vec{0}$.Comparing this with $x\vec{a} + y\vec{b} + z\vec{c} = \vec{0}$,we have $x = k-1$,$y = 1$,and $z = -k$.Summing these coefficients: $x + y + z = (k-1) + 1 + (-k) = 0$.

If $x\vec{a} + y\vec{b} + z\vec{c} = \vec{0}$,then under what condition are the three points $A, B, C$ with position vectors $\vec{a}, \vec{b}, \vec{c}$ collinear?

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