Vectors and vector addition:
A scalar is a quantity like mass or temperature that only has a magnitude. On the other had, a vector is a mathematical object that has magnitude and direction. A line of given length and pointing along a given direction, such as an arrow, is the typical representation of a vector. Typical notation to designate a vector is a boldfaced character, a character with and arrow on it, or a character with a line under it (i.e., ). The magnitude of a vector is its length and is normally denoted by or A. Addition of two vectors is accomplished by laying the vectors head to tail in sequence to create a triangle such as is shown in the figure. The following rules apply in vector algebra.where P and Q are vectors and a is a scalar.
Unit vectors:
A unit vector is a vector of unit length. A unit vector is sometimes denoted by replacing the arrow on a vector with a "^" or just adding a "^" on a boldfaced character (i.e., ). Therefore, Any vector can be made into a unit vector by dividing it by its length. Any vector can be fully represented by providing its magnitude and a unit vector along its direction.
Base vectors and vector components:
Base vectors are a set of vectors selected as a base to represent all other vectors. The idea is to construct each vector from the addition of vectors along the base directions. For example, the vector in the figure can be written as the sum of the three vectors u1, u2, and u3, each along the direction of one of the base vectors e1, e2, and e3, so that Each one of the vectors u1, u2, and u3 is parallel to one of the base vectors and can be written as scalar multiple of that base. Let u1, u2, and u3 denote these scalar multipliers such that one has<span> </span><span>The original vector</span><span> </span><span>u</span><span> </span><span>can now be written as </span><span>The scalar multipliers</span><span> </span><span>u</span><span>1</span><span>,</span><span> </span><span>u</span><span>2</span><span>, and</span><span> </span><span>u</span><span>3</span><span> </span><span>are known as the components of</span><span> </span><span>u</span><span> </span><span>in the base described by the base vectors</span><span> </span><span>e</span><span>1</span><span>,</span><span> </span><span>e</span><span>2</span><span>, and</span><span> </span><span>e</span><span>3</span><span>. If the base vectors are unit vectors, then the components represent the lengths, respectively, of the three vectors</span><span> </span><span>u</span><span>1</span><span>,</span><span> </span><span>u</span><span>2</span><span>, and</span><span> </span><span>u</span><span>3</span><span>. If the base vectors are unit vectors and are mutually orthogonal, then the base is known as an orthonormal, Euclidean, or Cartesian base.</span>
A vector can be resolved along any two directions in a plane containing it. The figure shows how the parallelogram rule is used to construct vectors a and b that add up to c. <span>In three dimensions, a vector can be resolved along any three non-coplanar lines. The figure shows how a vector can be resolved along the three directions by first finding a vector in the plane of two of the directions and then resolving this new vector along the two directions in the plane. </span><span>When vectors are represented in terms of base vectors and components, addition of two vectors results in the addition of the components of the vectors.</span>