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OUTLINE : 1. INTRODUCING VECTORS. 1.1 Scalars. 1.2 Vectors. 1.3 Unit vectors. 1.4 Vector algebra. 1.5 Simple examples. 1.1 Scalars. A scalar is a quantity with magnitude but no direction, any mathematical entity that can be represented by a number. Examples: Mass, temperature, energy, charge ...
Three numbers are needed to represent the magnitude and direction of a vector quantity in a three dimensional space. These quantities are called vector quantities. Vector quantities also satisfy two distinct operations, vector addition and multiplication of a vector by a scalar.
Example. Write the polar unit vectors r and θ in terms of the Cartesian unit vectors x and y . Unit Vectors. We are familiar with the unit vectors in Cartesian coordinates, where . points in the x-direction and . y-direction.
At this stage it is convenient to introduce unit vectors along each of the coordinate axes. Let xˆ be a vector of unit magnitude pointing in the positive x-direction, yˆ, a vector of unit magnitude in the positive y-direction, and zˆ a vector of unit magnitude in the positive z-direction.
Our basic unit types (dimensions) are length (L), time (T) and mass (M). When we do dimensional analysis we focus on the units of a physics equation without worrying about the numerical values.
Example. Find a unit vector in the direction of the vector ~v= (6,8,10). Dot Product Definition. The dot product of 2 vectors~a= (a 1,a 2,a 3) and~b= (b 1,b 2,b 3) in 3-dimensional space is defined to be the scalar ~a.~b= a 1b 1 +a 2b 2 +a 3b 3. This is the algebraic definition of the dot product
The force acting on an object is a vector. The direction of the vector specifies the line of action of the force, and the magnitude specifies how large the force is. Other examples of vectors include position; acceleration; electric field; electric current flow; heat flow; the normal to a surface.
