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Vector OperationsChapter 3 Section 2A + B = ?ABVector DimensionsWhen diagramming the motion of an object, with vectors, the direction and magnitude is described in x- and y- coordinates simultaneously.This allows vectors to be used for 1-d and 2-d motion.How can I get to the red dot starting from the origin and can only travel in a straight line? xyxyThere are 3 main ways that I can travel to get from the origin to the red dot by only traveling in a straight lines.

There are other ways as well and will be discussed in class.Solving For The Resultant of 2 Perpendicular VectorsWhen two vectors are perpendicular to each other it forms a right triangle, when the resultant is formed.Right triangles have special properties that can be used to solve specific parts of the triangle.Such as the length of sides and angles.Magnitude of a VectorTo determine the magnitude of two vectors, the Pythagorean Theorem can be usedAs long as the vectors are perpendicular to each other.

Pythagorean Theoremc=a+b(length of hypotenuse)=(length of leg)+(length of other leg)Applied Pythagorean Theorem

c2=a2+b2 R=y+x (Mathematics) (Physics)

R = Resultant; x = Change in x direction; y = Change in y direction abcxyRDirection of a VectoroppadjApplied Tangent Functiona=oppb=adjcxyRxyRxyR=Recall Vector PropertiesExample Problem A soldier travels due east for 350 meters then turns due north and travels for another 100 meters. What is the soldiers total displacement?Example PictureExample WorkExample AnswerR=364 m @ 15.95Vector ComponentsEvery vector can be broken down into its x and y components regardless of its magnitude or direction.Vectors Pointing Along a Single AxisWhen a vector points along a single axis, the second component of motion is equal to zero.

Vectors That Are Not Vertical or HorizontalAsk yourself these questions.How much of the vector projects onto the x-axis?How much of the vector projects onto the y-axis?

Components of a VectorxyAA xA xResolving Vectors into ComponentsComponents of a vector The projection of a vector along the axis of a coordinate system.x-component is parallel to the x-axisy-component is parallel to the y-axisThese components can either be positive or negative magnitudes.Any vector can be completely described by a set of perpendicular components.Vector Component EquationsExample ProblemBreak the following vector into its x- and y- components.A = 6.0 m/s @ 39Example Problem WorkExample Problem AnswerAx = 4.66 m/sAy = 3.78 m/sExample Problem:A plane takes off from the ground at an angle of 15 degrees from the horizontal with a velocity of 150mi/hr. What is the horizontal and vertical velocity of the plane?

Example PictureExample WorkExample AnswerHorizontal velocity = 144.89 miles per hourVx=144.89mi/hr

Vertical velocity = 38.82 miles per hourVy=38.82mi/hrAdding Non-Perpendicular VectorsWhen vectors are not perpendicular, the tangent function and Pythagorean Theorem cant be used to find the resultant.Pythagorean Theorem and Tangent only work for two vectors that are at 90 degrees (right angles)Non-Perpendicular VectorsTo determine the magnitude and direction of the resultant of two or more non-perpendicular vectors:Break each of the vectors into its x- and y- components.It is best to setup a table to nicely organize your components for each vector.Component Tablex-componenty-componentVector A - (A)Vector B - (B)Vector C - (C)Add more rows if neededResultant - (R)Non-Perpendicular VectorsOnce each vector is broken into its x- and y- components :The components along each axis can be added together to find the resultant vectors components.Rx = Ax + Bx + Cx + Ry = Ay + By + Cy + Only then can the Pythagorean Theorem and Tangent function can be used to find the Resultants magnitude and direction.

Example ProblemDuring a rodeo, a clown runs 8.0m north, turns 35 degrees east of north, and runs 3.5m. Then after waiting for the bull to come near, the clown turns due east and runs 5.0m to exit the arena. What is the clowns total displacement?Practice Problem PictureStep #1: Draw a picture of the problemPractice problem WorkStep #2: Break each vector into its x- and y- components.x-componenty-componentVector A - (A)Vector B - (B)Vector C - (C)Resultant - (R)Step #3: Find the resultants components by adding the components along the x- and y-axis.x-componenty-componentVector A - (A)Vector B - (B)Vector C - (C)Resultant - (R)+Step #4: Find the magnitude of the vector by using the Pythagorean theorem.R2 = x2 + y2Step #5: Find the direction of the vector by using the tangent function.Tan = y/xStep #5: Complete the final answer for the resultant with its magnitude and direction.Practice Problem AnswerResultant displacement = 12.92m @ 57.21