Vectors - Scalars and Vectors

Jumat, 07 Februari 2020

Scalars and Vectors

A vector is a quantity having both a magnitude and a direction. Examples of vector quantities are velocity, force and position. One can represent a vector in n-dimensional space with an arrow whose initial point is at the origin, (Figure 2.1). The magnitude is the length of the vector. Typographically, variables representing vectors are often written in capital letters, bold face or with a vector over-line, A, a,. The magnitude of a vector is denoted |a|. A scalar has only a magnitude. Examples of scalar quantities are mass, time and speed.

Vector Algebra.  Two vectors are equal if they have the same magnitude and direction. The negative of a vector, denoted −a, is a vector of the same magnitude as a but in the opposite direction. We add two vectors a and b by placing the tail of b at the head of a and defining a + b to be the vector with tail at the origin and head at the head of b. (See Figure 2.2.).

The difference, a − b, is defined as the sum of a and the negative of b, a + (−b). The result of multiplying a by a scalar α is a vector of magnitude |α||a| with the same/opposite direction if α is positive/negative. (See Figure 2.2.) Figure 2.1:

Figure 2.1: Graphical representation of a vector in three dimensions.

Figure 2.2: Vector arithmetic.

Here are the properties of adding vectors and multiplying them by a scalar. They are evident from geometric considerations.

a + b = b + a                            αa = aα                         commutative laws

(a + b) + c = a + (b + c)           α(βa) = (αβ)a               associative laws

α(a + b) = αa + αb                   (α + β)a = αa + βa        distributive laws

Zero and Unit Vectors. The additive identity element for vectors is the zero vector or null vector. This is a vector of magnitude zero which is denoted as 0. A unit vector is a vector of magnitude one. If a is nonzero then a/|a| is a unit vector in the direction of a. Unit vectors are often denoted with a caret over-line, .

Rectangular Unit Vectors. In n dimensional Cartesian space, Rⁿ, the unit vectors in the directions of the coordinates axes are e₁,...e_n. These are called the rectangular unit vectors. To cut down on subscripts, the unit vectors in three dimensional space are often denoted with i, j and k. (Figure 2.3).

Figure 2.3: Rectangular unit vectors.

Components of a Vector. Consider a vector a with tail at the origin and head having the Cartesian coordinates (a₁,...,a_n). We can represent this vector as the sum of n rectangular component vectors, a = a₁e₁ +···+ a_ne_n. (See Figure 2.4.) Another notation for the vector a is. By the Pythagorean theorem, the magnitude of the vector a is.

Figure 2.4: Components of a vector.

Sumber

Labels: Mathematician

Thanks for reading Vectors - Scalars and Vectors. Please share...!