scalar product example

Scalar (mathematics) - In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector. More generally, a vector space may be defined by using any field instead of the real numbers, such as the complex numbers. Then the scalars of that vector space will be the elements of the associated field. Also, a scalar product operation (not to be confused with scalar multiplication) may be defined on a vector space, allowing two vectors to be multiplied to produce a scalar. A vector space equipped with a scalar product is called an inner product space. The real component of a quaternion is also called its scalar part. The term is also sometimes used informally to mean a vector, matrix, tensor, or other usually 'compound' value that is actually reduced to a single component. Thus, for example, the product of a 1× n matrix and an n ×1 matrix, which..

Scalar multiplication - In mathematics, scalar multiplication, also known as multiplication, is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). Note that scalar multiplication is different from scalar product which is an inner product..

  Physics 240 Lecture 6 - Scalar Product and Cartesian Coordinate SystemProfessor Jerzy Wrobel explains introduces scalar product and shows how to identify components of vectors.

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Question : I'm proving the formula for the area of a triangle using vectors, and I am told that the Area '@' = 1/2*|AB x AC| = 1/2*|BA x BC| = 1*2|CA x CB| . Where A,B,C are the points of the triangle. My question is, couldn't I use the scalar product in these formulae and get the same result? As we're only taking the magnitude into consideration, I don't see why the answer could not be '1*2|AB . AC|'. Cheers, Andrew

Answer : The formula is 1/2 * product of lengths of two sides * sin(angle between the sides) Since there is sin of the angle, therefore we use cross product. __________.

Question : Do you get a vector or a scalar product when you... 1) Add/subtract two scalars ? 2) Multiply/divide two scalars ? 3) Add/subtract two vectors ? 4) Multiply/divide two vectors ? 5) Add/subtract a vector and scalar ? 6) Multiply/divide a vector and scalar ?

Answer : 1,3) Adding or subtracting a scalar or vector gives the same. 2) Multiplication/Division of scalar gives scalar. 4) There are two types of product: one is the DOT(.) and other is CROSS(X) The DOT product gives a scalar while CROSS gives another vector perpendicular to the plane formed by the two vectors. 6) Multiply/Divide of a scalar and vector always gives a vector.