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Chapter 3 Vector Algebra. 《Advanced Algebra and Analytic Geometry》 subject development group. 3.1 Vectors and Linear Operations. I. Vectors. 1. Vector. Vector is the one that has both size and direction. Such as velocity, acceleration, force and displacement, etc.
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Chapter 3 Vector Algebra 《Advanced Algebra and Analytic Geometry》 subject development group
I. Vectors 1. Vector. Vector is the one that has both size and direction. Such as velocity, acceleration, force and displacement, etc. 2. Vector representation. Vector is denoted by directed line segment. The length of the directed line segment denote the vectorial size, and the direction of the directed line segment denote the vectorial direction. Example Vector can also be denoted by a letter. Such as or
3. Moduleof vector.The size of vector is called module of vector, which is denoted by . The vector that has module 1 is called identity vector. The vector that has module 0 is called zero vector, which is denoted by or 0. The initial point and terminal point of zero vector are coincident, while the direction is arbitrary. 4. Two equivalent vectors. and are equivalent if they have the same size and the uniform direction, denoted by
5. Free vector. The vector that is independent of initial point is called free vector (vector). 6. Opposite vector. If two vectors and have the same size but the opposite direction, we call is the opposite vector of , denoted by . 7. Zero vector. A vector is called zero vector if its length is 0, denoted by 0. 8. Identity vector. Identity vector is the vector with length 1.
1. Addition and subtraction of vectors Parallelogram law. Construct vectors by way of a point , then construct a parallelogram such that are the edges of it, is called sum vector of the two vectors, denoted by the foregoing exposition is called the parallelogram law of vector addition. II. Linear operations of vector How to comprehend the sum of n vectors?
(2) addition is associative Operation law of vector addition. (1) addition is commutative
Deference of two vectors. It’s not difficult to get.
2. Multiplication of vector and number Product rule of vector and number. The product of vector andnumber is a vector denoted by , and If the direction of the vector and is same (opposite). Specially, Operation law. (1) associative law
(2) distributive law III. Collinear Vectors and Coplanar Vectors Definition Vectors are called collinear vectors if their direction are same or opposite. While vectors that parallel with the same plane are called coplanar vectors.
Theorem 1 Let vector thenthe necessary and sufficientcondition of is that their exists an unique such that Proof It’s only need to prove necessity because sufficiency isobvious. Let find a real number satisfying is a posotive number if and have the same direction, and is a negative number if theirdirection are opposite,
then Uniqueness. Let we get namely, is obvious for Theorem 2 The necessary and sufficient condition of coplanarfor threevectors is that there exist three numbers (not all equal zero) such that
are collinear such as Proof If two vectors in by Theorem 3.1, there exist that not all equal zeroes are not all equal zeroes, such that then we have Assume that are all non-collinear, construct construct a straight line by point that parallel with and intersect a line that
lies in at point by triangle law and the definition of scalar product, we have then where are not all equal zeroes. Contrarily, if there exist that not all equal zeroessuch that
then It’s might as well assume It shows that is a diagonal line ofparallelogram by the edge of are coplanar. Hence
Example 3.1 For given is the midpoint of edge prove Proof. By triangle law we have
and for is the midpoint of so Example 3.2 Prove the Median Line Theorem by vector. Example 3.3 Prove by vector. If is barycenter
of and is mean line of the edge then
