The Vector Lecture Or Whats Our Vector Victor

The Vector Lecture Or What’s Our Vector Victor?

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

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

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.

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 Angle R cosine angle R

Use of Velocity Vectors: Wind and Wave Power • 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 Wind A B C

Vector vs. Scalar Quantities Scalars Vectors • Have magnitude only • Add algebraically • Examples • Magnitude and direction • Add geometrically • Examples – – – Time Mass Energy Temperature Heat and Internal Energy • Represented with normal face type w/o line above value – – – 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

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

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

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

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

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 Qx Qy Py Q P

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

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

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

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