Vectors Chapter 6Vectors Chapter 6

KONICHEKKONICHEK

JUST DOING SOME ANGLING

TERMINOLGY

I.Vector vrs Scalar A.has magnitude and direction B. Scalar- measures only magnitude or direction( speed)

II.Resultant vector B."sum" of several vectors; the effect of the resultant vector is

always greater than the effect of its individual components. III.Concurrent vectors

A.act on the same point at the same time IVEquilibrant vector

A.a vector that produces equilibrium; it is equal in magnitude and opposite in direction to the resultant vector Graphical method of vector addition:

HOW VECTOR DIAGRAMS WORKHOW VECTOR DIAGRAMS WORK

V.Graphical method of vector addition:V.Graphical method of vector addition: A.Vectors are represented graphically by A.Vectors are represented graphically by using arrows. using arrows. 1.The length of the arrow represents the 1.The length of the arrow represents the vector’s magnitudevector’s magnitude 2.the direction the arrow points represents 2.the direction the arrow points represents the direction of the vector. the direction of the vector. 3.Vectors are drawn graphically using a 3.Vectors are drawn graphically using a scale. scale.

B.Vectors are added graphically by placing them "tips to tails." B.Vectors are added graphically by placing them "tips to tails." 1.The tail of the second vector is touches the tip of the first 1.The tail of the second vector is touches the tip of the first vector, etc. vector, etc. 2.The resultant vector is drawn graphically by placing the tail 2.The resultant vector is drawn graphically by placing the tail of the resultant at the tail of the first vector and the tip of the of the resultant at the tail of the first vector and the tip of the resultant at the tip of the last vector. resultant at the tip of the last vector. 3.It is drawn from where you started to where you ended. 3.It is drawn from where you started to where you ended. 4.The resultant vector’s magnitude can be determined 4.The resultant vector’s magnitude can be determined graphically by measuring its length and converting its length using graphically by measuring its length and converting its length using the scale chosen. the scale chosen. 5.Its direction is the direction that it points.5.Its direction is the direction that it points. 6.There are two parts that describe the resultant vector 6.There are two parts that describe the resultant vector graphically—its magnitude and its compass direction.graphically—its magnitude and its compass direction.

Vector addition

Mathematical method Mathematical method of vector additionof vector addition

I.One dimension: I.One dimension: A.Vectors that act in a line linearly are A.Vectors that act in a line linearly are

assigned positive and negative signs to assigned positive and negative signs to indicate their direction. Positive signs are indicate their direction. Positive signs are assigned to vectors acting right, up, east, assigned to vectors acting right, up, east, or north. Negative signs are assigned to or north. Negative signs are assigned to vectors acting left, down, west, or south. vectors acting left, down, west, or south.

B.Using their signs, vectors are added B.Using their signs, vectors are added algebraically to determine the magnitude algebraically to determine the magnitude and sign (direction) of the resultantand sign (direction) of the resultant

II.Two dimensions: II.Two dimensions: A.The resultant vector is found mathematically. We will A.The resultant vector is found mathematically. We will

use the component method of vector addition. A resultant use the component method of vector addition. A resultant vector can be considered to be the vector sum of its vector can be considered to be the vector sum of its resultant x-component and its resultant y-component, resultant x-component and its resultant y-component, separated by 90°. This can also be done using graphing separated by 90°. This can also be done using graphing calculators. calculators.

B.The magnitude of the resultant vector can be B.The magnitude of the resultant vector can be determined using the Pythagorean theorem determined using the Pythagorean theorem

1.c1.c22 = a = a22 + b + b22 (you remember this!!!) (you remember this!!!) The direction of the resultant vector can be expressed as an The direction of the resultant vector can be expressed as an

angle between 0° and 90°, and use the appropriate trig angle between 0° and 90°, and use the appropriate trig function function

a) sin =o/h, cos a/h , tan o/a ( this might be new)a) sin =o/h, cos a/h , tan o/a ( this might be new) 1. 1. right triangle trigonometry using SO/H-CA/H-TO/Aright triangle trigonometry using SO/H-CA/H-TO/A

Practice- sometimes drawing a Practice- sometimes drawing a picture helps visualizing( advised)picture helps visualizing( advised)

a boat is traveling at 7m/s across a a boat is traveling at 7m/s across a river. The current is 1.5m/s. find the river. The current is 1.5m/s. find the vector of the boat.vector of the boat.

Mag= 7Mag= 722 + 1.5 + 1.522 = = 7.2m/s7.2m/s Direction is: TanDirection is: Tan-1 -1 = 1.5/7= = 1.5/7= 12.1˚12.1˚

VECTOR COMPONENTSVECTOR COMPONENTS Let's find the size of the x-component; that is, Let's find the size of the x-component; that is, let's find the size of the adjacent side.let's find the size of the adjacent side.We know the hypotenuse, (316 Newtons), and we We know the hypotenuse, (316 Newtons), and we know the angle, (35 degrees). We want to find the know the angle, (35 degrees). We want to find the length of the adjacent side, (x-component). What length of the adjacent side, (x-component). What trigonometry function relates the hypotenuse, an trigonometry function relates the hypotenuse, an acute angle and its adjacent side in a right acute angle and its adjacent side in a right triangle? The cosine function does. The math looks triangle? The cosine function does. The math looks this way:this way: Now, since the original vector is named Now, since the original vector is named FF, its x-, its x-component is named Fx. This would be read 'F sub component is named Fx. This would be read 'F sub x'. So, in the above math we should remove 'x-x'. So, in the above math we should remove 'x-component' and replace that term with Fx, as in:component' and replace that term with Fx, as in: We can solve for Fx by doing a little algebra and We can solve for Fx by doing a little algebra and looking up the cosine of thirty-five degrees:looking up the cosine of thirty-five degrees: 35˚= F/316N35˚= F/316N