If P₁ , P₂, P₃, P₄ are points in a plane or space and O, the origin of vectors, show that P₄ coincides with O if an only if
$stack OP subscript 1 with rightwards arrow on top space plus space stack straight P subscript 1 straight P subscript 2 with rightwards arrow on top space plus space stack straight P subscript 2 straight P subscript 3 with rightwards arrow on top space plus space stack straight P subscript 3 straight P subscript 4 with rightwards arrow on top space equals space 0 with rightwards arrow on top.$

stack OP subscript 1 with rightwards arrow on top space plus space stack straight P subscript 1 straight P subscript 2 with rightwards arrow on top space plus space stack straight P subscript 2 straight P subscript 3 with rightwards arrow on top space equals space stack straight P subscript 3 straight P subscript 4 with rightwards arrow on top space equals space 0 with rightwards arrow on top.

left right double arrow space space space open parentheses stack OP subscript 1 space with rightwards arrow on top plus stack straight P subscript 1 straight P subscript 2 with rightwards arrow on top close parentheses space plus space open parentheses stack straight P subscript 2 straight P subscript 3 with rightwards arrow on top space plus space stack straight P subscript 3 straight P subscript 4 with rightwards arrow on top close parentheses space equals space 0 with rightwards arrow on top left right double arrow space space space stack OP subscript 2 with rightwards arrow on top space plus space stack straight P subscript 2 straight P subscript 4 with rightwards arrow on top space equals space 0 with rightwards arrow on top space space space space space left right double arrow space space space space space space space stack OP subscript 4 with rightwards arrow on top space equals space 0 with rightwards arrow on top space space space space left right double arrow space space straight O space and space straight P subscript 4 space coincide.

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Vector Algebra

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