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A vector is a quantity that has both a magnitude (or size) and a direction. Both of these properties must be given in order to specify a vector completely. In this unit we describe how to write down vectors, how to add and subtract them, and how to use them in geometry.
17 Σεπ 2022 · Definition \(\PageIndex{2}\): Length of a Vector. Let \(\vec{u} = \left[ u_{1} \cdots u_{n} \right]^T\) be a vector in \(\mathbb{R}^n\). Then, the length of \(\vec{u}\), written \(\| \vec{u} \|\) is given by \[\| \vec{u} \| = \sqrt{ u_{1}^2 + \cdots + u_{n}^2}\nonumber \]
A vector is a quantity that has both magnitude (size) and direction. Force & velocity are commonly used vectors in physics, but we focus on ‘displacement’ vectors in GCSE maths – these give the magnitude and direction of a movement from one point to another .
An example of a vector is a = [4, 3] . Graphically, you can think of this vector as an arrow in the x-y plane, pointing from the origin to the point at x=3, y=4 (see illustration.) In this example, the list of numbers was only two elements long, but in principle it could be any length. The dimensionality of a vector is the length of the list.
Example 1. Find a vectora with representation given by the directed line segment AB. −→. Draw AB and the equivalent representation starting at the origin. A(1, 2), B(3, 3); A(1, −2), B(−2, 3). The magnitude (length) |a | ofa is the length of any its representation. The length ofa =< a1, a2 > is. a | = qa2. + a2 2. −→.
Definition 13.1. a) A vector represents the length and direction of a line segment. The length is denoted V . A unit vector U is a vector of length 1. The direction of a vctor V is the unit vector U parallel to V: U V V . b) Given two points P Q, the vector from P to Q is denoted PQ. c) Addition.
Basic Vector Algebra in 1. Vector Equality: Two vectors and are equal if and only if and . 2. Vector Addition: The sum of the vectors and is defined by. 3. Scalar Multiplication: Suppose is a vector and . Then the scalar product of is defined by. Example Find the sum of the following vectors. 1. , 2. , 3. ,
