5 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
6 Adding vectors Graphically When you have many vectors, just keep repeating the process until all are includedThe resultant is still drawn from the origin of the first vector to the end of the last vector.
7 Alternative Graphical Method When you have only two vectors, you may use the Parallelogram MethodAll vectors, including the resultant, are drawn from a common origin
8 Properties of Vector addition Vectors obey the Commutative Law of AdditionThe order in which the vectors are added does not affect the result
9 Properties of Vector addition Vectors also obey the Associativity Law of AdditionWhen adding three vectors, it does not matter which two yo start with
10 Scalar Multiplication of Vectors Associative lawDistributive law
15 Vector Decomposition Vector is decomposed to vectors and . is the projection of the vector along the x-axisyxis the projection of the vector along the x-axisVector is decomposed to vectors and .Vector is the sum of its components:How do we find and ?
16 Unit vectors y Both and vectors x Both and vectorsThe magnitude of the unit vectors equals 1:yxThe vector is expressed as
18 Unit vector in 3D cartesian coordinates Unit vector in the directions of vector
19 Adding and subtracting vectors Algebraic method
20 Dot product of vectorsDot product (or scalar product) of vectors and is defined asDot product is always a scalar quantityTwo vectors are orthogonal (i.e. perpendicular to each other) if their dot product is zero
22 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 asCross product is always a vector perpendicular to the plane.
27 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 asFind the magnitude of a vector if the resultant ofand is in the y-axis and its magnitude is 5.2.
28 Dot products 1. What is the angle between and 2. If what is the angle between them?