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# S_notes02 - 11—1 PARALLELOGRAM FACTS IN E3 Facts The vertices of a parallelogram must be coplanar Opposite sides of a parallelogram must have

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Unformatted text preview: 11—1 PARALLELOGRAM FACTS IN E3 Facts: The vertices of a parallelogram must be coplanar. Opposite sides of a parallelogram must have equal length. If a convex quadrilateral has a pair of opposite sides which are parallel and of equal length, then it must be a parallelogram. ARROWS AND VECTORS Deﬁnition: A line segment whose ends are designated as its initial and ﬁnal points ( “tail” point and “head” point) is called an arrow. An arrow of positive length determines a unique line. Two arrows of positive length are said to be parallel if their unique lines are parallel. Fact: Let P and Q be the tail and head of a given arrow PQ, and let P' any point; then must be a unique arrow P’Q' such that PQ and P'Q’ have the same length and are parallel, and hence such that the ﬁgure PQQ'P' is a parallelogram. In this case, we say that the two arrows are equal. Let PQ be a given arrow of positive length. The set of all possible arrows in E3 which are equal to PQ is called a vector or, more speciﬁcally, the vector determined by the arrow PQ. Given the arrow PQ, or any arrow equal to PQ, we say that this arrow represents this vector. Thus it is possible for the same vector to be represented by several different arrows, provided that those arrows are all equal to one another. TL? PARALLELOGRAM FACTS IN E3 Facts: The vertices of a parallelogram must be coplanar. Opposite sides of a parallelogram must have equal length. lfa convex quadrilateral has a pair of opposite sides which are parallel and of equal length, then it must be a parallelogram. ARROWS AND VECTORS Deﬁnition: A line segment whose ends are designated as its points {“tail” point “head” polar) is called an mw. An arrow of positive length determines a unique line. Two arrows ofpositive length are said to be parallel if their unique lines are parallel. Fact: Let P and Q be the tail and head ofa given arrow PQ, and let P’ any point; then must be a unique arrow P’Q’ such that PQ and P’Q’ have the same length and are parallel, and hence such that the ﬁgure PQQ'P’ is a parallelogram. In this case, we say that the two arrows are equal. Let PQ be a given arrow of positive length. The set of all possible arrows in E5 which are equal to PQ is called a vector or, more speciﬁcally, the vector determined by the arrow sz. Given the arrow PQ, or any arrow equal to PQ, we say that this arrow represents this vector. Thus it is possible for the same vector to be represented by several different arrows, provided that those arrows are all equal to one another. ...
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## This note was uploaded on 09/24/2011 for the course MATH 1802 taught by Professor Duorg during the Two '04 term at Macquarie.

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S_notes02 - 11—1 PARALLELOGRAM FACTS IN E3 Facts The vertices of a parallelogram must be coplanar Opposite sides of a parallelogram must have

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