The magnitude (length) of a vector is the square root of the sum of the squares of its elements. If you visualize a vector as a directed magnitude, running from the origin to the specified point in n-space, then you can think of adding vectors as the geometric equivalent of placing the vectors "tail to head". In particular, this graphical way of thinking about scalar multiplication of a vector remains true for vectors of any dimensionality. Vectors of higher dimensionality can't really be visualized, but the 2-D or 3-D intuitions are often useful in thinking about higher-dimensional cases as well. We can visualize this easily in 2-space (where points are defined by pairs of numbers), and also in 3-space (where points are defined by triples of numbers). For any (non-zero) vector, any point on the line can be reached by some scalar multiplication. Multiplying the vector's elements by a scalar moves the point along that line. A vector - viewed as a point in space - defines a line drawn through the origin and that point. Scalar multiplication has a simple geometric interpretation. You can't add two vectors of different sizes. number of elements) can be added: this adds the corresponding elements to create a new vector of the same size. Operations on vectors include scalar multiplication (also of course scalar addition, subtraction, division)Īnd vector addition. The individual numbers that make up a vector are called elements or components of the vector. You can enter a vector in Matlab by surrounding a sequence of numbers with open and close square brackets: We'll try to avoid uses whose meaning is not clear from context. the number of elements in it (so that a vector has length 2 in this sense), or we might mean the geometric distance from the origin to the point denoted by the vector (so that a vector has length 5 in this sense). Note 2: ordinary language words such as length or size can be ambiguous when applied to a vector: we might mean the dimensionality of the vector, i.e. Note 1: the origin is the vector consisting of all zeros - in whatever dimensionality. The dimensionality n can be anything: 1 or 37 or 10 million or whatever. Geometrically, a vector of dimensionality n can be interpreted as point in an n-dimensional space, or as a directed magnitude (running from the origin to that point). a member of R^n (where R stands for the real numbers). Operations defined on scalars include addition, multiplication, exponentiation, etc.Ī vector is an n-tuple (an ordered set) of numbers, e.g. The name "scalar" derives from its role in scaling vectors. COGS501 - Homework 1 part 1 - Matlab tour Vectors, matrices and basic operations on themĪ scalar is any real (later complex) number.
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