General Physics 101 PHYS Dr. Zyad Ahmed Tawfik Email : zmohammed@inaya.edu.sa Website : zyadinaya.wordpress.com

Lecture No.1 Vectors Vectors

VECTORS

Objectives Define scalar and vector quantity. Addition of vectors using geometrical method. Addition of vectors using Pythagorean Theorem Addition of vectors using analytical methods.

All Physical quantities can be divided into two types, 1. Scalar quantities and 2.Vector quantities

Scalar A Scalar is any quantity has Magnitude, but No direction. Magnitude – is a numerical value with units. Scalar Example Magnitude length 3 m Distance 10 m Age 15 years Temperature 40 degree

Vector A Vector is any quantity has BOTH Magnitude and Direction. Magnitude & Direction Velocity 20 m/s, North Motion 3 m/s, South Force 5 N, Up

SCALAR VECTOR distance volume speed displacement acceleration mass work power resistance force velocity weight pressure

A vector is represented by an arrow A vector is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector. Blue and orange vectors have same magnitude but different direction. Blue and purple vectors have same magnitude and same direction so they are equal. Blue and green vectors have same direction but different magnitude.

Kinds of Vectors

1) Zero vector or Null vector: A vector having zero magnitude is called a Null vector or Zero vector. 2) Equal vectors: Vectors are said to be equal if both vectors have same magnitude and direction A=B A B 3) Parallel vectors (Like vectors): Vectors are said to be parallel if they have the same directions. A The vectors A and B represent parallel vectors. B Note: Two equal vectors are always parallel but, two parallel vectors may or may not be equal vectors

The vectors A and B are anti parallel vectors 4) Anti parallel vectors (Unlike vectors): Vectors are said to be anti parallel if they acts in opposite direction. The vectors A and B are anti parallel vectors 5) Negative vector : The negative vector of any vector is a vector having equal magnitude but acts in opposite direction A . B Negative of a vector is just a vector going the opposite direction . A A = - B or B = - A B 6) Orthogonal vectors: Two vectors are said to be orthogonal to one another if the angle between them is 90°. B A

Subtraction of Vectors

VECTOR SUBTRACTION VECTOR SUBTRACTION - If 2 vectors are going in opposite directions, you SUBTRACT. Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started? 54.5 m, E - 30 m, W 24.5 m, E

Example, A bear, searching for food walked 30 meters east, then 10 meters west. Calculate the bear's displacement. 30 m, E - 10 m, W Solution : Since, A = 30 m, E & B = 10 m, W Then, C = A + B = A + (- B) = A – B = 30 - 10 = 20 m, E

Resultant of Two Vectors The resultant is the sum or the combined effect of two vector quantities Vectors in the same direction: 6 N 4 N = 10 N 6 m = 10 m 4 m Vectors in opposite directions: 6 m s-1 10 m s-1 = 4 m s-1 6 N 10 N = 4 N

If 2 similar vectors point in the SAME direction, add them. Example: A man walks 50 meters east, then another 30 meters east. Calculate total his displacement. + 50 m, E 30 m, E 80 m, E A = 50 + 30 = 80 m, E

Addition of Vectors

Adding vector There are three methods to adding Vector 1- Graphical or called (Geometrical Method) 2- Pythagorean Theorem 3- Analytical Method or called Component's Method

Geometric Method

In this method you need the technical tools like sharp pencil, ruler, protractor and the paper (graphing or bond) to show the vectors graphically. 1) Add vectors A and B graphically by drawing them together in a head to tail arrangement. 2) Draw vector A first, and then draw vector B such that its tail is on the head of vector A. 3) Then draw the sum, or resultant vector, by drawing a vector from the tail of A to the head of B. 4) Measure the magnitude and direction of the resultant vector

In this method To add vectors, we put the start point of the second vector on the end point of the first vector. The resultant vector is distance between start point and end point . End point Start point

In the previous example, DISPLACEMENT was asked for and since it is a VECTOR we should include a DIRECTION on our final answer. N W of N E of N N of E N of W W E N of E S of W S of E NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HEAD TO TOE. W of S E of S S

Example 1 A man walks at40 meters East and 30 meters North. Find the magnitude of resultant displacement and its vector angle. Use Graphical Method.

Solution: Write the given facts A = 40 meters East B = 30 meters North R = ? θ = ?

graph the vectors from the origin (head to tail) NOTE: 1 GRID = 10 METERS USE RULER TO MEASURE AND TO DRAW A LINE R = 50 METERS B = 30 METERS, NORTH θ = 37° N of E graph the vectors from the origin (head to tail) A = 40 METERS, EAST

Example2 2. Given: A = 5 km ,East B = 6 km, NE C = 7 km, 30˚ N of W R = ? θ = ? f

graph the vectors from the origin (head to tail) θ = 37° N of E 2. Given: A = 5 km ,East B = 6 km, NE C = 7 km, 30˚ N of W R = ? θ = ? NOTE: 1 GRID = 10 km POSSIBLE GRAPH USE RULER TO MEASURE AND TO DRAW A LINE R = ? B = 30 METERS, NORTH graph the vectors from the origin (head to tail) A = 40 METERS, EAST

Pythagorean Theorem

The Pythagorean Theorem The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other. The method is not applicable for adding more than two vectors or for adding vectors that are not at 90-degrees to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.

Example1 A man walks at 40 meters East and 30 meters North. Find the magnitude of resultant displacement and its vector angle. Use Pythagorean Theorem.

θ R = 50 METERS sketch your problem B = 30 METERS, NORTH     sketch your problem R = 50 METERS B = 30 METERS, NORTH θ A = 40 METERS, EAST

Component method

The component method is one way to add vectors. The component method of vector addition is the standard way to add vectors

In this method Each vector has two components : the x-component and the y-component If the vectors are in secondary directions : (NW, NE, SW or SE directions) Ax = A cos θx Ay = A sin θx where: A = the given vector value θx = the given angle from x -axis Ax = the x – component of vector A Ay = y – component of vector A

Vectors Components The figure shows a vector A and an xy-coordinate system. We can define two new vectors parallel to the x and y axes, named the component vectors of A, Slide 3-30

NEED A VALUE OF ANGLE! To find a numeric value for the angle, we used the following laws. Hypotenuse Opposite q Adjacent

Or, Cos θ = b / c Sin θ = a / c Tan θ = a/b c a q b

Also: cos θ = Ax/A Or Ax = A cos θ . & sin θ = Ay/A Or Ay = A sin θ Ay θ Ax

Example 1 A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement. 23 m, E - = 12 m, W - = 14 m, N 6 m, S 20 m, N 35 m, E R 14 m, N q 23 m, E The Final Answer: 26.93 m, 31.3 degrees NORTH or EAST

Example 2 Find the components of the vector A , If A = 2 and the angle θ = 30o (cos 30 = 0.866 & sin 30 = 0.500). Ay Ax θ Solution, Since, Ax = A Cos θ Then, Ax = 2 cos 30 = 2 x 0.866 = 1.73 Also, Ay = A sin 30 = 2 x 0.5 = 1

Example 3 plane moves with a velocity of 63 Example 3 plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate the plane's horizontal and vertical velocity components. H.C. =? 32 V.C. = ? 63.5 m/s

Example 4: A boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to north direction. 8 m/s, W 15 m/s, N C q Then, C = 17 m/s with 28.1o , NW

Example 5: if the magnitude of A is A=2, θ = 30o , and the magnitude of B is B=3 and θ = 70o find the magnitude of A +b ? θAB= 70-30=40

Example 6: A=2x+y and b=4x+7y a)find the components of C=A+B b) find the magnitude of C and its angle θ with x axis ? a) C=A+B=2x+y+4x+7y= 6x+8y Thus the components of C=A+B is Cx= 6 and Cy= 8 b) The magnitude of C By using Pythagorean theorem

QUESTIONS: 1.What is a scalar? 2.What is a vector 3.Identify whether the following quantities are scalars or vectors? (i) Mass (ii) weight (iii)speed (iv)velocity (v)energy (vi)work (vii)force (ix)temperature (x)pressure (xi)angular momentum (xii)wavelength. 4. what does it mean: Null vector - Negative vector - Equal vector – Parallel vector - Anti parallel vector ?

4. A vector has magnitude of 5 units and direction angle qA=300 4. A vector has magnitude of 5 units and direction angle qA=300 . Find Ax and Ay? 5. A storm system moves 5000 km due east, then shifts course at 40 degrees North of East for 1500 km. Calculate the storm's resultant displacement 6. Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal and vertical components?

7. if the magnitude of A is A=5, θ = 20o , and the magnitude of B is B=7 and θ = 80o find the magnitude of A +B ? 8. the components of a vector are defined by Ax =3.46 and Ay =2 find the magnitude and direction angle of the vector A ?

1- Graphical method 2- Pythagorean Theorem. 8). A man walks at 60 meters East and 20 meters North. Find the magnitude of resultant displacement and its vector angle use 1- Graphical method 2- Pythagorean Theorem.

Thank You for your Attention