Vector Algebra and Operations in Engineering Mathematics, Lecture notes of Engineering Science and Technology

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MAT 247 Engineering Mathematics

Hakkı Ula¸s Unal¨ Dept. of Electrical-Electronics Eng. Anadolu University, Turkey

September 26, 2018

Today

(^1) Vector Algebra

Vectors

Physical quantities are often specified by one number or magnitude. The physical quantities, such a mass, time, temperature, speed are fully specified by one number, which we call scalar. However, they don’t provide us more information. For instance, speed of a car does not say anything about its direction.

Vectors

Physical quantities are often specified by one number or magnitude. The physical quantities, such a mass, time, temperature, speed are fully specified by one number, which we call scalar. However, they don’t provide us more information. For instance, speed of a car does not say anything about its direction. There exists some quantities, such as force, velocity, which require more than one number to describe them. A vector is a quantity that has both magnitude and direction. We can say that a vector is an arrow or a directed line segment.

Component of vectors

Dimension of a vector corresponds to number of its elements. A method of representing a vector is to list its elements in a sufficient number of different directions, which define a coordinate frame.

Component of vectors

Dimension of a vector corresponds to number of its elements. A method of representing a vector is to list its elements in a sufficient number of different directions, which define a coordinate frame. In this course, we will focus on Cartesian coordinate frame

Vector Equality

A vector a is equal to a vector b means, they have same magnitude (or length) and a direction. This means a = [a 1 , a 2 , a 3 ] and b = [b 1 , b 2 , b 3 ] in a Cartesian coordinate frame are equal to each other iff a 1 = b 1 a 2 = b 2 a 3 = b 3

Vector Equality

A vector a is equal to a vector b means, they have same magnitude (or length) and a direction. This means a = [a 1 , a 2 , a 3 ] and b = [b 1 , b 2 , b 3 ] in a Cartesian coordinate frame are equal to each other iff a 1 = b 1 a 2 = b 2 a 3 = b 3 Magnitude of a vector a is

a = |a| =

a^21 + a^22 + a^23.

Vector Summation and Substraction

Let a = [a 1 , a 2 , a 3 ] and b = [b 1 , b 2 , b 3 ]. Then,

a + b = [a 1 + b 1 , a 2 + b 2 , a 3 + b 3 ,

and a − b = a + (−b) = [a 1 − b 1 , a 2 − b 2 , a 3 − b 3 ,

Vector Summation and Substraction

Let a = [a 1 , a 2 , a 3 ] and b = [b 1 , b 2 , b 3 ]. Then,

a + b = [a 1 + b 1 , a 2 + b 2 , a 3 + b 3 ,

and a − b = a + (−b) = [a 1 − b 1 , a 2 − b 2 , a 3 − b 3 , Properties of vector summation: I (^) a + b = b + a I (^) (a + b) + c = a + (b + c) = a + b + c I (^) a + 0 = a I (^) a + (−a) = 0

Scalar Multiplication of Vector

Multiplication of a vector a by a scalar c (c is real number) can be defined as ca = [ca 1 , ca 2 , ca 3 ]. Scalar multiplication keeps the direction of the vector a if c > 0, otherwise, the direction is reversed unless c = 0.

Scalar Multiplication of Vector

Multiplication of a vector a by a scalar c (c is real number) can be defined as ca = [ca 1 , ca 2 , ca 3 ]. Scalar multiplication keeps the direction of the vector a if c > 0, otherwise, the direction is reversed unless c = 0. Properties of scalar multiplication: I (^) c(a + b) = ca + cb I (^) c(ka) = (ck)a I (^1) a = a

Dot or Inner Product

Inner product of two vectors a = [a 1 , a 2 , a 3 ] and b = [b 1 , b 2 , b 3 ] is defined as: a·b = a 1 b 1 + a 2 b 2 + a 3 b 3 ] Properties of inner product: a·b = b · a a · (b + c) = a · b + a · c (ka) · b = k(a · b)

Geometric Interpretation of Inner Product

a

c b

θ

c^2 = a^2 + b^2 − 2 ab cos θ

|a − b|^2 = (a − b) · (a − b) |a − b|^2 = a · a + b · b − 2(a · b) |a − b|^2 = a^2 + b^2 − 2(a · b)