General Physics (PHYS101) Vector analysis Lecture 06 Chapter 3

Outline Addition and subtraction of vectors Vector decomposition Unit vectors Dot (scalar) product of vectors Cross (vector) product of vectors

Adding vectors When adding vectors, their directions must be taken into account. Units must be the same. Graphical methods Use scale drawings Algebraic methods More convenient

Adding vectors Graphically triangle method y x

Adding vectors Graphically Continue drawing the vectors “tip-to-tail” The resultant vector is drawn from the origin of the first vector to the and of the second one. Measure the length of the resultant vector and its angle

Adding vectors Graphically When you have many vectors, just keep repeating the process until all are included The resultant is still drawn from the origin of the first vector to the end of the last vector.

Alternative Graphical Method When you have only two vectors, you may use the Parallelogram Method All vectors, including the resultant, are drawn from a common origin

Properties of Vector addition Vectors obey the Commutative Law of Addition The order in which the vectors are added does not affect the result

Properties of Vector addition Vectors also obey the Associativity Law of Addition When adding three vectors, it does not matter which two yo start with

Scalar Multiplication of Vectors Associative law Distributive law

Vector Subtraction y x

Vector Subtraction Special case of vector addition If A-B, then use A+B: Continue with standard vector addition procedure

Vector Subtraction y x 𝑫=𝑨−𝑩

General Physics (PHYS101) Golibjon Berdiyorov Building 6, Room 148 Vector analysis Lecture 06 Chapter 3

Vector Decomposition Vector is decomposed to vectors and . is the projection of the vector along the x-axis y x is the projection of the vector along the x-axis Vector is decomposed to vectors and . Vector is the sum of its components: How do we find and ?

Unit vectors y Both and vectors x Both and vectors The magnitude of the unit vectors equals 1: y x The vector is expressed as

Unit vectors y x The vector is expressed as y x y x

Unit vector in 3D cartesian coordinates Unit vector in the directions of vector

Adding and subtracting vectors Algebraic method

Dot product of vectors Dot product (or scalar product) of vectors and is defined as Dot product is always a scalar quantity Two vectors are orthogonal (i.e. perpendicular to each other) if their dot product is zero

Dot products

Cross product of vectors Cross product is a vector operation that generates a new vector from the other two vectors. The magnitude of cross product of vectors and is defined as Cross product is always a vector perpendicular to the plane.

Properties of cross product The cross product is anti-commutative since changing the order of the vectors cross product changes the direction of the resulting vector

Mathematical definition of cross product Two vectors are parallel to each other if and only if:

Cross products

Vector analysis 1. The angle between and the negative y-axis is? 2. A vector in the xy plane has a magnitude of 25 m and an x component of +12 m and a positive y component. Find the vector? The angle it makes with the positive y axis is? 3. If has the magnitude of 3 m and makes an angle 30o with the +x axis, then the vector is? 4. 5. A vector is defined as Find the magnitude of a vector if the resultant of and is in the y-axis and its magnitude is 5.2.

Dot products 1. What is the angle between and 2. If what is the angle between them?

Cross products