2 Scalar SCLR is NY quantity in physics that has MGNITUDE, but NOT a direction associated with it. Magnitude numerical value with units. Scalar Example Speed Distance ge Magnitude 20 m/s 10 m 15 years Heat 1000 calories
3 Vector VECTOR is NY quantity in physics that has OTH MGNITUDE and DIRECTION. r v, r x, r a, r F Vector Velocity cceleration Force Magnitude & Direction 20 m/s, N 10 m/s/s, E 5 N, West Vectors are typically illustrated by drawing an RROW above the symbol. The arrow is used to convey direction and magnitude.
4 Polar Notation Polar notation defines a vector by designating the vector s magnitude and angle θ relative to the +x axis. Using that notation the vector is written: In this picture we have a force vector with magnitude 12 Newtons oriented at 210 degrees with the + x axis. It would be characterized as F 12 < 210
5 Polar Notation In this picture we have a force vector of 12 Newtons oriented along the -x axis. However, polar notation is relative to the + x axis. Therefore, it would be characterized by F 12 < 180 In this last picture we have 2 vectors. They are characterized by: C 2 < 30 D 4 < - 50 or D 4 < 310
6 Scalar Multiplication Multiplying a vector by -1 does not change the magnitude, but it does reverse it's direction or in a sense, it's angle. Multiplying a vector by a scalar will ONLY CHNGE its magnitude. Thus if 12 < 105, Then 2 24 < 105-1/2 Thus if 12 < 105, then - 12 < 285 If 12 < 105, then (-1/2) 6 < 285
7 Unit Vector Notation n effective and popular system used in engineering is called unit vector notation. It is used to denote vectors with an x-y Cartesian coordinate system.
8 Unit Vector Notation J vector of magnitude 1 in the y direction 3j i vector of magnitude 1 in the x direction 4i The hypotenuse in Physics is called the RESULTNT or VECTOR SUM. 4 iˆ + 3 ˆj 3j The LEGS of the triangle are called the COMPONENTS Vertical Component Horizontal Component 4i NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components HED TO TOE.
9 Unit Vector Notation iˆ ˆj kˆ - unit vector 1in the + x direction - unit vector 1in the + y direction - unit vector 1in the + z direction The proper terminology is to use the hat instead of the arrow. So we have i-hat, j-hat, and k-hat which are used to describe any type of motion in 3D space. How would you write vectors J and K in unit vector notation? J 2 iˆ + 4 ˆj K 2iˆ 5 ˆj
10 pplications of Vectors VECTOR DDITION If 2 similar vectors point in the SME direction, add them. Example: man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started? 54.5 m, E + 30 m, E 84.5 m, E Notice that the SIZE of the arrow conveys MGNITUDE and the way it was drawn conveys DIRECTION.
11 pplications of Vectors VECTOR SUTRCTION - If 2 vectors are going in opposite directions, you SUTRCT. Example: man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started? 24.5 m, E 54.5 m, E 30 m, W -
12 Non-Collinear Vectors When 2 vectors are perpendicular, you must use the Pythagorean theorem. man walks 95 km, East then 55 km, north. Calculate his RESULTNT DISPLCEMENT. c 2 a 2 + b 2 c a 2 + b 2 55 km, N c Resultant c km 95 km,e
13 UT..what about the VLUE of the angle??? Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VLUE of the angle km 55 km, N To find the value of the angle we use a Trig function called TNGENT. θ N of E 95 km,e opposite Tanθ adjacent θ Tan 1 (0.5789) side side o So the COMPLETE final answer is : < 30 95ˆ i m + 55 ˆj m o NofE
14 What if you are missing a component? Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal and vertical components? H.C.? The goal: LWYS MKE RIGHT TRINGLE! V.C? m To solve for components, we often use the trig functions sine and cosine. cosineθ adj hyp cosθ adjacent side hypotenuse sineθ opp opposite side hypotenuse hypsinθ adj V. C. 65cos m, N or 58.91ˆj m opp H. C. 65sin m, E or 27.47ˆ i m
15 Example 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 m, E 12 m, W 6 m, S 20 m, N 35 m, E R - 14 m, N 14 m, N R Tanθ θ Tan (0.6087) m o 31.3 The Final nswer: θ 23 m, E < ˆ i m + 14 ˆj m o NofE
16 Example 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 due north. R v 8.0 m/s, W θ 15 m/s, N R v m / s 8 Tanθ θ Tan (0.5333) 2 o 28.1 The Final nswer : 17 m 17 m o / 28.1 WofN o / s < ˆ i m / s + 15 ˆj m / s
17 Example 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.? cosineθ adj hyp cosθ adjacent side hypotenuse sineθ opp opposite side hypotenuse hypsinθ m/s V.C.? adj H. C. 63.5cos m / s, E or 53.85ˆ i opp V. C. 63.5sin m / s, S or ˆj
18 The Dot Product (Vector Multiplication) Multiplying 2 vectors sometimes gives you a SCLR quantity which we call the SCLR DOT PRODUCT. In polar notation consider 2 vectors: < θ 1 & < θ 2 The dot product between and produces a SCLR quantity. The magnitude of the scalar product is defined as: Where θ is the NET angle between the two vectors. s shown in the figure.
19 The Scalar Dot Product Let 12 < 30, Let 5 < 65 What is "dot"? cosθ 12 5 cos35 In unit vector notation, it looks a little different. Consider: The "Dot" product between these is equal to:
20 The Scalar Dot Product What is the SIGNIFICNCE of the dot product?
21 The significance of the dot product In this figure, vector has been split into 2 components, one PRLLEL to vector and one PERPENDICULR to vector. Notice that the component parallel to vector has a magnitude of Cosθ THEREFORE when you find the DOT PRODUCT, the result is: i) The MGNITUDE of one vector, in this case and, ii) The MGNITUDE of the 2nd vector's component that runs parallel to the first vector. (That is where the cosine comes from)
22 Dot Products in Physics Consider this situation: force F is applied to a moving object as it transverses over a frictionless surface for a displacement, d. s F is applied to the object it will increase the object's speed! ut which part of F really causes the object to increase in speed? It is F Cosθ! ecause it is parallel to the displacement d In fact if you apply the dot product, you get ( F Cosθ)d, which happens to be defined as "WORK" (check your equation sheet!) W F x cosθ F x cosθ Work is a type of energy and energy DOES NOT have a direction, that is why WORK is a scalar or in this case a SCLR PRODUCT (K DOT PRODUCT).
23 Dot products using the TI-89 Here is where having the TI-89 is going to come if handy. You can actually enter vector into your calculator in UNIT VECTOR NOTTION. On your calculator we first need to clear any data that might be lurking in its memory. Go to F6, then choose CLER a-z. This will wipe out any and all stored values that you have assigned to variable letters. Suppose we had two vectors, one which represented the force and the other which represents the displacement as in our WORK example earlier. Now let's learn how to type a vector into the calculator in unit vector notation.
24 Dot products using the TI-89 You start off by using brackets, then separate each number by a comma, then close the brackets. Then hit the STO button, then "a" as shown below in the diagram. Then do the same thing for the next vector except store this as "b" Now that you have assigned each vector a variable. On the calculator go to CTLOG and find dotp( from the menu). Hit enter. Then type in a,b and close the parenthesis as shown in the diagrams.
25 Dot products using the TI-89 We see we get -1 for the WORK. What does this mean? Remember, the dot product gives you a SCLR which has NO DIRECTION. The negative here simply means that the Force and Displacement oppose each other yet are still PRLLEL. n example would be the WORK done by friction to slow a base runner down. Force of Friction Displacement
26 The Cross Product (Vector Multiplication) Multiplying 2 vectors sometimes gives you a VECTOR quantity which we call the VECTOR CROSS PRODUCT. In polar notation consider 2 vectors: < θ 1 & < θ 2 The cross product between and produces a VECTOR quantity. The magnitude of the vector product is defined as: θ Where θ is the NET angle between the two vectors. s shown in the figure.
27 The Vector Cross Product θ 30kˆ sinθ 12 5 sin150 What about the direction???? Positive k-hat??? We can use what is called the RIGHT HND THUM RULE. Fingers are the first vector, Palm is the second vector, Thumb is the direction of the cross product. Cross your fingers,, towards, so that they CURL. The direction it moves will be either clockwise (NEGTIVE) or counter clockwise (POSITIVE) In our example, the thumb points OUTWRD which is the Z axis and thus our answer would be 30 k-hat since the curl moves counter clockwise.
28 Cross Products and Unit Vectors The cross product between and produces a VECTOR of which a 3x3 matrix is need to evaluate the magnitude and direction. iˆ x x ˆj y y kˆ z z You start by making a 3x3 matrix with 3 columns, one for i, j, & k-hat. The components then go under each appropriate column. Since is the first vector it comes first in the matrix
29 Cross Products and Unit Vectors z y x z y x k j i ˆ ˆ ˆ ) ( ) ( ˆ y z z y i ) ( ) ( ˆ z x x z j You then make an X in the columns OTHER THN the unit vectors you are working with. For i, cross j x k For j, cross i x k For k, cross i x j Let s start with the i-hat vector: We cross j x k ) ( ) ( ˆ x y y x k Now the j-hat vector: We cross i x k Now the k-hat vector: We cross i x j
30 Example Let s start with the i-hat vector: We cross j x k iˆ ( 4)(5) ( 6)( 4) 44 Now the j-hat vector: We cross i x k ˆj ( 6)(3) ( 2)(5) 8 Now the k-hat vector: We cross i x j k ˆ ( 2)( 4) ( 4)(3) 20 The final answer would be: 44 iˆ 8 ˆj + 20kˆ
31 The significance of the cross product In this figure, vector has been split into 2 components, one PRLLEL to vector and one PERPENDICULR to vector. Notice that the component perpendicular to vector has a magnitude of sinθ THEREFORE when you find the CROSS PRODUCT, the result is: i) The MGNITUDE of one vector, in this case and, ii) The MGNITUDE of the 2nd vector's component that runs perpendicular to the first vector. ( that is where the sine comes from)
32 Cross Products in Physics There are many cross products in physics. You will see the matrix system when you learn to analyze circuits with multiple batteries. The cross product system will also be used in mechanics (rotation) as well as understanding the behavior of particles in magnetic fields. force F is applied to a wrench a displacement r from a specific point of rotation (ie. a bolt). Common sense will tell us the larger r is the easier it will be to turn the bolt. ut which part of F actually causes the wrench to turn? F Sin θ r r F F r sinθ sinθ
33 Cross Products in Physics r r F F sinθ r sinθ What about the DIRECTION? Which way will the wrench turn? Counter Clockwise Is the turning direction positive or negative? Positive Which way will the OLT move? IN or OUT of the page? OUT You have to remember that cross products give you a direction on the OTHER axis from the 2 you are crossing. So if r is on the x-axis and F is on the y-axis, the cross products direction is on the z-axis. In this case, a POSITIVE k-hat.
34 Cross products using the TI-89 Let s clear the stored variables in the TI-89 r 0.30m F 5N? Since r is ONLY on the x-axis, it ONLY has an i-hat value. Enter, zeros for the other unit vectors. Using CTLOG, find crossp from the menu. Cross x, then x Look at the answers carefully! Crossing x shows that we have a direction on the z-axis. Since it is negative it rotates CW. This mean the OLT is being tightened or moves IN to the page. Crossing x, causes the bolt to loosen as it moves OUT of the page
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