There is a vector 0 such that b + 0 = b (additive identity) įor any vector a, there is a vector − a such that a + (− a) = 0 (Additive inverse). Given a vector a and a real number (scalar) λ, we can form the vector λ a as follows. If λ is positive, then λ a is the vector whose direction is the same as the direction of a and whose length is λ times the length of a. In this case, multiplication by λ simply stretches (if λ>1) or compresses (if 0<λ<1) the vector a. If, on the other hand, λ is negative, then we have to take the opposite of a before stretching or compressing it. In other words, the vector λ a points in the opposite direction of a, and the length of λ a is |λ| times the length of a. No matter the sign of λ, we observe that the magnitude of λ a is |λ| times the magnitude of a: ∥λ a∥ = |λ| ∥ a∥.
(λ + β) a = λ a + β b (distributive law for scalars) Λ( a + b) = λ a + λ b (distributive law, for vectors) Scalar multiplications satisfies many of the same properties as the usual multiplication. In the last formula, the zero on the left is the scalar 0, while the zero on the right is the vector 0, which is the unique vector whose length is zero. If a = λ b for some scalar λ, then we say that the vectors a and b are parallel.
If λ is negative, then it is a common slang to say that a and b are anti-parallel, but we will not use that language. Generalizing well-known examples of vectors (velocity and force) in physics and engineering, mathematicians introduced abstract object called vectors. So vectors are objects that can be added/subtracted and multiplied by scalars. These two operations (internal addition and external scalar multiplication) are assumed to satisfy natural conditions described above.Ī set of vectors is said to form a vector space (also called a linear space), if any vectors from it can be added/subtracted and multiplied by scalars, subject to regular properties of addition and multiplication. Wind, for example, has both a speed and a direction and, Hence, is conveniently expressed as a vector. The first thing we need to know is how to define a vector so it (Weight is the force produced by the acceleration of gravity acting on a mass.) The same can be said of moving objects, momentum, forces, electromagnetic fields, and weight. Today more than ever, information technologies are an integral #Wolfram mathematica 7 tutorial how to# #Wolfram mathematica 7 tutorial how to#.
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