1-dimensional: origin. Above is a vector drawing that represents the displacement of the point from zero. point on a line: x = 2

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Marianna Hardy
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1 I. WHT IS VECTO? UNIT XX: VECTOS VECTO is a variable quantity consisting of two components: o o MGNITUDE: How big? This can represent length, pressure, rate, and other quantities DIECTION: Which way is the magnitude pointed or exerted? vector is represented symbolically with an arrow, or in equations as a letter with an arrow over it,. MGNITUDE Let s start thinking about vectors by thinking about the displacement of a point from its origin. The magnitude of the DISPLCEMENT a of a point tells how far (length) it is from a starting point or origin. In physics, this is NOT the same as DISTNCE. origin 1-dimensional: point on a line: x = 2 bove is a vector drawing that represents the displacement of the point from zero How would you describe the magnitude of this vector? 1

It is easy to see that the magnitude of the vector above is 2 units. 1-dimensional vectors will fall parallel to either the x-axis (horizontal) or the y-axis (vertical).

2 It is easy to see that the magnitude of the vector above is 2 units. 1-dimensional vectors will fall parallel to either the x-axis (horizontal) or the y-axis (vertical). y 2-dimensional y origin x x point in a plane: x = 3, y = -2 vector that describes the displacement of point (3, -2) from the origin How would you define the magnitude of this vector? 2

The magnitude of the vector above can be found by measuring the length from the origin to the point (for example, using a ruler or graph paper), or by using the distance formula: (x 2 x 1 ) 2 + (y 2

We say that the vector has OTH an x-component and a y-component.

3 The magnitude of the vector above can be found by measuring the length from the origin to the point (for example, using a ruler or graph paper), or by using the distance formula: (x 2 x 1 ) 2 + (y 2 y 1 ) 2 = (3 0) 2 + ( 2 0) 2 = = 13 units In a 2-dimensional plane, the vector length must account for displacement along TWO axes. We say that the vector has OTH an x-component and a y-component. You you like, you can think of the 1-dimenional vectors as 2-dimensional vectors that have either an x-component = 0 (vertical) or y-component = 0 (horizontal) y x vector with x-component = 0 vector with y-component = 0 y x vector with x and y components ll the x and y-component stuff means is that we can say how big a vector is in the x direction and in the y direction. This will be important again when we discuss FOCES! 3

The x and y components of a 2-dimensional vector (x 2, y 2 ) y x the length of this dotted line is the magnitude of the y-component of the vector y Note the dotted line is parallel to an imaginary

4 The x and y components of a 2-dimensional vector (x 2, y 2 ) y x the length of this dotted line is the magnitude of the y-component of the vector y Note the dotted line is parallel to an imaginary y-axis (x 1, y 1 ) x the length of this dotted line is the magnitude of the x-component of the vector x = x 2 x 1 y = y 2 y 1 = x 2 + y 2 Note the dotted line is parallel to an imaginary x-axis ESOLVING THE X ND Y COMPONENTS OF VECTO LWYS YIELDS IGHT TINGLE. THEEFOE, WE CN USE PYTHGOS THEOEM TO COMPUTE THE MGNITUDE OF THE VECTO. 4

5 . DIECTION Direction can be described in a number of different ways. We could indicate a + or direction, we could use compass directions like east, north-west, etc., we could even say up and to the right. For 2-dimensional vectors, it is common to define an angle with reference to the x-axis (East) in an x-y plane. The angle of a vector is defined as the angle formed where the vector and an imaginary x-axis meet, measured in a counter-clockwise direction. Examples: y vector angle = x vector angle = 35 0 ngles of 0 o or are NOT the same since the direc;on is opposite lso, angles of 90 0 and 270 o are not the same. EMEME: VECTO IS DESCIED Y OTH MGNITUDE ND DIECTION 5

6 DIECTION MTTES!!! = If the magnitude of these two PLLEL vectors is equal, are the two vectors the same? Does? What would we get if they were added? 6

7 II. DDING VECTOS Here s the anatomy of a vector: tail-no arrow direction = angle with x-axis (East), CCW length = magnitude head-arrow end When vectors are added, you are finding what is called the ESULTNT vector. Vectors are always added by placing them head-to-tail. DDING 1-DIMENSIONL VECTOS DD: 5 units, 0 o + = 3 units, units, 0 0 o + = 8 units, 0 These vectors have the same direction. When I put them together, head to tail, the resultant vector will be 8 units long, 0 o net direction (due East) 7

8 DD: 5 units, 0 o + = 3 units, units, 0 0 o + = 2 units, 0 vectors laid head to tail resultant is the sum These vectors have the opposite direction. When I put them together, head to tail, the resultant vector will be 2 units long, 0 o net direction asically, because the vectors are in opposite directions, their magnitudes are opposite in sign. When added, the resultant has to be smaller. If vector had been larger than vector, in which direction would the resultant point?. DDING 2-DIMENSIONL VECTOS Let s start with two vectors that form a 90 o angle when added The vectors to be added are always placed head-to-tail The resultant vector is formed by connecting the tail of the first vector to the head of the last vector Pythagorean Theorem is used to compute the magnitude of the resultant The direction of the resultant is found using sin, cos, or tan relationships 8

9 Here we are adding two vectors with the same magnitude but directions that form a 90 o angle when added. head to tail 90 o angle sides = 11 units θ This is super easy to do because vector only has a y-component and vector only has an x-component. Can you see, then, that one dimensional vectors are just the x or y component of their resultant? DIECTION sinθ = opp hyp = = θ = sin = 44.8 o The direction is 45 o which is what you d expect from a triangle with the length of both sides being equal. MGNITUDE a 2 + b 2 = = 242 = 15.6 km This is just an application of the Pythagorean theorem. 9

10 DD: + = 6 units, 0 o 4 units, 270 o vectors placed head to tail MGNITUDE a 2 + b 2 = = 52 = 7.21 units DIECTION θ The direction is the angle the resultant makes with an imaginary x-axis (East), measured counter clockwise. Can you see the direction angle will be 270 o + θ = o tanθ = opp adj = 6 4 = 1.5 θ = tan = 56.3 o 10

11 Can you see that when two vectors are added that form a 90 o angle, the horizontal vector is just the x-component of the resultant and the vertical vector is just the y-component of the resultant? C. dding two vectors that do not form a 90 o angle when added How would you add: = 5 units,70 o = 3 units, 30 o Place them head to tail, without disturbing their directions. The order you place them makes no difference (adding vectors obeys the commutative and associative properties) These two diagrams are the same 11

12 oth vectors and have x and y components. The sum of the x-components of and equal the x-component of the resultant. The sum of the y-components of and equal the y-component of the resultant. y y y x x cos 70 o = adj hyp = x 5 x = 5cos 70 o = 1.71 units sin 70 o = opp hyp = y 5 y = 5sin 70 o = 4.70 units cos 30 o = adj hyp = x 3 x = 3cos 30 o = 2.59 units sin 30 o = opp hyp = y 3 y = 3sin 30 o = 1.5 units x x = x + x = 4.30 units y = y + y = 6.20 units = 2 x + 2 y = 7.54 units magnitude of the resultant 12

13 y x MGNITUDE OF THE ESULTNT x = x + x = 4.30 units y = y + y = 6.20 units = 2 x + 2 y = 7.54 units DIECTION OF THE ESULTNT tanθ = opp adj = y x = = 1.44 θ = tan = 55.3 o The ESULTNT vector has a magnitude of 7.54 units and a direction of 55.3 o 13

14 D. dding more than 2 vectors This can be done graphically, which we will do in class. This can be done with math, just like we did in the last section Let s add: = 10 units,50 o = 4 units,120 o C = 4.5 units, 310 o dd them head to tail (order doesn t matter). Connect the tail of the first vector with the head of the last one drawn. That s the resultant. C Each of these vectors has an x and y component. We re going to do what we did before add up the x components and y components to get the resultant 14

15 C sin50 o = opp hyp = y 10 y = 5sin 50 o = 7.66 units C cos50 o = adj hyp = x 10 x = 10cos50 o = 6.43 units cos120 o = adj hyp = x 4 x = 4 cos120 o = 2 units cos 310 o = adj hyp = x 4.5 x = 4.5 cos 310 o = 2.89 units sin120 o = opp hyp = y 4 y = 4 sin120 o = 3.46 units sin 310 o = opp hyp = y 4.5 y = 4.5sin 310 o = 3.45 units For the resultant, just add up the x and y components and use Pythagoras (like we did for 2 vector addition) 15

16 y = y + y + C x = 7.66 units = MGNITUDE x 2 + y 2 = units x = x + x + C x = 7.32 units DIECTION tanθ = opp adj = y x = = 1.04 θ = tan = 46.1 o Let s apply some common sense Look at the vector drawing again. Does it make sense that vector and C will essentially cancel each other? Go look at the values of the x and y components again on the previous page. Does it makes sense, therefore, that the resultant and vector should have similar magnitude and direction? oth the drawing and the math tell you the same thing! C 16

Objectives and Essential Questions

VECTORS Objectives and Essential Questions Objectives Distinguish between basic trigonometric functions (SOH CAH TOA) Distinguish between vector and scalar quantities Add vectors using graphical and analytical