The Vector Lecture

1. The Vector Lecture Or What’s Our Vector Victor?

2. Vectors vs. Scalar Quantities • Vector quantities have both a magnitude (measure) and direction. • Vector quantities add graphically • Vector quantities have components usually perpendicular to one another. • Scalar quantities have direction only • Scalar quantities add algebraically (simply add values) • No components • Can have positive and negative values. • THE TEST

3. Examples • Displacement, velocity and acceleration • Force • Momentum and Impulse • Torque, angular acceleration, angular velocity and angular displacement • Time, mass • Energy and Work • Rotational kinetic energy

4. Vector Addition • For vectors in same or opposite direction simply add or subtract. • For vectors at right angles use pythagorus to add and get resultant. • For vectors at angles other than right angle break up vector(s) into components, add and then use pythagorus. • Graphically can always use tip to tail and using scaling for magnitude.

5. Vector Decomposition • Dividing one vector into two component vectors • Components are amount of vector quantity in that direction • Perpendicular components do not affect one another • Parallel Component = R cosine angle • Perpendicular Component = R sine angle • Opposite of vector addition R sine angle R Angle R cosine angle

6. Use of Velocity Vectors: Wind and Wave Power Wind • Fastest point of sail is nearly perpendicular to wind • Running with wind means you can only go as fast as the wind • Running sideways to the wind means that wind continues to push on you even when you are going the same speed as wind A B C

7. Vector vs. Scalar Quantities Scalars Vectors Magnitude and direction Add geometrically Examples Displacement Velocity Acceleration Force Momentum Torque Represented using arrows where length is magnitude and direction is direction Represented in text with boldface or with line above value • Have magnitude only • Add algebraically • Examples • Time • Mass • Energy • Temperature • Heat and Internal Energy • Represented with normal face type w/o line above value

8. Check Question 2 Which of the following represents the resulting vector for the two vectors to the right? C B A

9. Check Question 1 Which of the pictured vectors are in the same direction? • A and B • B and C • A and C Which of the pictured vectors are equal in magnitude? • A and B • B and C • A and C A B C

10. Vector Addition • Vector addition is when two vectors are added to become one resultant vector • To add vectors they must be of the same measurement type and unit. • Graphically add vectors by using tip to tail or parallelogram method • Mathematically adding vectors • Simply add magnitudes if in same direction • Subtract magnitudes if in opposite direction • Use Pythagorean Theorem if at right angles • Direction found using tan q = y/x where y and x are vertical and horizontal measurement of same type

11. Check Question 3 • Show the perpendicular components of the following resultant vectors. A B D C

12. Vector Addition Revisited • To add vectors not in same direction or at right angles. • Decompose each vector into perpendicular components • Add components in same direction • Find resultant using Pythagorean theorem • Find angle using tangent or other trig function and components • This is where the law of sines and cosines came from Px Py Qx P Q Qy

13. Use of Force Vectors: Inclined Plane • Part of the weight tries to push mass into plane while part of weight tries to pull mass down plane • As the angle of the incline plane goes from horizontal to vertical • The amount of the weight into the plane decreases • The amount of the weight down the plane increases • The overall weight stays the same

14. A Use of Force Vectors: Mechanical Equilibrium • Equilibrium • Forces in horizontal direction add to equal zero • Forces in vertical direction add to equal zero • Motion is that of constant velocity (model 1) • one possibility is the object remains stationary • NO ACCELERATION • Examples • String breaks when ends of string are pulled apart • Unsupported chain cannot be pulled straight • Bridge Demo Variables

15. Representing Vector Quantities • Text book uses Bold type • Can also use arrow over value • Arrow representation • Length gives magnitude • Direction gives direction

16. Vector Quantities • Vector quantities are those having magnitude and direction. Non-vector quantities are scalar. • A vector quantity partially goes in one direction and partially in another • The test: does it make sense if you put directions after it? • Examples: velocity, acceleration, displacement, force, momentum and torque • Vector quantities add differently than scalar quantities