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## Chapter 3

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**Chapter 3**Vectors Herriman High AP Physics C**Frame of Reference**• In order to describe motion, you have to pick a “frame of reference” • Frame of reference is always decided by the observer • In physics we often use a coordinate axis to denote our frame of reference. Herriman High AP Physics C**Frame of Reference**Y -x x This is a familiar coordinate axis, but in physics it has a slightly different meaning. -Y Herriman High AP Physics C**Frame of Reference**• In this frame of reference, up is positive, right is positive • Down in negative, as is left • Hence in physics a negative has no value, it merely denotes direction • This is necessary because direction has a special distinction in physics; it separates vectorquantities fromscalarquantities Herriman High AP Physics C**Vector vs. Scalar**• A Scalar Quantity is one that has only magnitude – distance is a scalar • Example: If you travel 500 miles that is a distance • A Vector Quantity is one that has magnitude and direction – displacement is a vector • Example: If you travel 500 miles North this is displacement Herriman High AP Physics C**Vector vs. Scalar**• If you divide distance by time you get average speed • Example: S = D/t = 500 miles/2 hours =250 mph • If you divide displacement by time you get average velocity • Example: Vavg = x/t = 500 miles North/2 hours = 250 mph North Herriman High AP Physics C**Vector Addition**• Two Methods • Graphical Method • Requires a ruler and a protractor • Process • Convert each vector into a line that fits a scale of your choosing • Draw Vectors head to tail, measuring the angle exactly • Draw a resultant • Measure the resultant and the resulting angle • Convert the measurement back into a vector answer Herriman High AP Physics C**Graphical Addition**• A man drives 125 km west and then turns 45º North of West and travels an additional 100 km. What is his displacement? • Step one: Set Scale • 1 cm = 25 Km 100 km = 4 cm 45° Herriman High AP Physics C 125 km = 5 cm**Graphical Addition**Step Two: Move Vectors Head to Tail Step Three: Draw Resultant from Tail of First Vector to the head of the last. 100 km = 4 cm 45° 125 km = 5 cm Herriman High AP Physics C**Graphical Addition**Step Four: Measure the Measure the Resultant and the resulting angle. Step Five: Using the Scale, convert the measurement into an answer. 100 km = 4 cm 8.3 cm 8.3 cm * 25 Km/cm = 208 Km 45° 125 km = 5 cm 19.9° Final Answer: His displacement is 208 Km @ 19.9° North of West Herriman High AP Physics C**Mathematical Addition**• Mathematical Addition of Vectors Requires a basic knowledge of Geometry – You must know: • Sin θ = Opposite/Hypotenuse • Cos θ = Adjacent/Hypotenuse • Tan θ = Sin θ/ Cos θ =Opposite/Adjacent • ArcSin, ArcCos, ArcTan Herriman High AP Physics C**Mathematical Addition**• Mathematical addition requires that you be able to • Draw a rough sketch of the original vectors • Resolve each vector into its Components • Reduce the component vectors into a system of two vectors • Draw a parallelogram • Draw the Resultant • Use Triangle geometry to find the magnitude and direction of the resultant Herriman High AP Physics C**Mathematical Addition**Step One: Draw Original Vectors 45º 125 Km West 100 Km 45° North of West Herriman High AP Physics C**Mathematical Addition**Step Two: Resolve Vectors into components Cos θ = W/100 km W = 100 Cos 45° =70.7 Km Sin θ = N/100 km N = 100 Sin 45° =70.7 Km 125 Km West 100 Km 45° North of West Herriman High AP Physics C**Mathematical Addition**Step Three: Reduce to a system of two vectors 195.7 Km West 70.7 Km North 70.7 Km North 70.7 Km West = + 125 Km West Herriman High AP Physics C**B**A Mathematical Addition • Step 4: Draw Parallelogram • Step 5: Draw Resultant • Step 6: Use Right Triangle Geometry Pythagorean Theorem: C2 = A2 + B2 C = SQRT(A2 + B2) = SQRT{(197.5)2+(70.7)2} = 209.8 km Tan θ = 70.7/197.5 = 0.3579 Θ= ArcTan 0.3579 = 19.7° North or West Herriman High AP Physics C**Graphical vs. Mathematical Methods**• Graphical Method • Less Complicated Math • Good Approximation • Not as exact as you would like • Mathematical Method • No need for ruler or protractor • Less time dedicated to sketch • More Accurate result • Mathematics is more difficult Herriman High AP Physics C**Component Vectors Unit Vectors**• The unit vector in the x direction is denoted by i • The unit vector in the y direction is denoted by j • The unit vector in the z direction is denoted by k Rule: i2 = j2 = k2 = 1 Herriman High AP Physics C**B**A Component Vectors Unit Vectors • In terms of component unit vectors: • Vector A =Cx= 5i • Vector B = Cy = 3j • Hence Vector C = 5i + 3j 5.83 units 3 units C 5 units Herriman High AP Physics C**Adding Vectors Component Method**• Vector A = 2i + 3j - 2k • Vector B = 5i- 6j + 8k • A + B = 7i – 3j + 6k Magnitude of the Vector |A + B| = SQRT{(7i)2+(-3j)2+(6k)2} = 9.7 Rules of Vector Addition • A + B = B + A • A – B = A + (-B) Herriman High AP Physics C**Multiplication of Vectors**• Three types of Vector Multiplication • Multiply a vector by a scalar • Vector A = 3i + 2j + 5k • Vector 4A = 12i + 8j + 20k Herriman High AP Physics C**Multiplication of Vectors**• Multiply 2 Vectors to get a scalar (dot product) • a•b = |a||b| cos θ = axbx+ ayby + azbz Rule: i•i=j•j = k•k = 1 i•j = i•k = j•k = 0 • Vector A = 2i + 3j - 2k • Vector B = 5i- 6j + 8k • a•b =(2i*5i) + (3j * -6j) + (-2k * 8k) = 10 – 18 – 16 = -24 Herriman High AP Physics C**Multiplication of Vectors**• Multiply 2 Vectors to get a Vector (cross product) • A x b = |a||b| sin θ = axby+ axbz + aybx + aybz + azbx+ azby Rule: i x i=j x j = k x k = 0 i x j = k ; j x k = i ; k x i = j j x i = -k ; k x j = -i ; i x k = -j Herriman High AP Physics C**Vector A = 2i + 3j - 2k**Vector B = 5i- 6j + 8k A x B = (2i x -6j) + (2i x 8k) + (3j x 5i) + (3j x 8k) + (-2k x 5i) + (-2k x -6j) -12k + - 16j + -15k + 24i + -10j + -12i 12i - 26j – 27k Multiplication of Vectors Herriman High AP Physics C