2 Vector 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.
3 How can I get to the red dot starting from the origin and can only travel in a straight line? xy
4 There are 3 main different ways that I can travel to get from the origin to the red dot by only traveling in a straight lines.xy
5 Solving For The Resultant of 2 Perpendicular Vectors When 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.
6 (length of hypotenuse)²=(length of leg)²+(length of other leg)² 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 Theoremc²=a²+b²(length of hypotenuse)²=(length of leg)²+(length of other leg)²
19 Resolving Vectors into Components Components 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.
20 Vector Component Equations Solving for the x-component of a vector.𝐴𝑥=𝐴𝑐𝑜𝑠𝜃(𝑥−𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴 = 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴 • cos(𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴)Solving for the y-component of a vector.𝐴𝑦=𝐴𝑠𝑖𝑛𝜃
21 Example ProblemBreak the following vector into its x- and y- components.A = °
27 Example Answer Horizontal velocity = 144.89 miles per hour Vx=144.89mi/hrVertical velocity = miles per hourVy=38.82mi/hr
28 Adding Non-Perpendicular Vectors When vectors are not perpendicular, the tangent function and Pythagorean Theorem can’t be used to find the resultant.Pythagorean Theorem and Tangent only work for two vectors that are at 90 degrees (right angles)
29 Non-Perpendicular Vectors To determine the magnitude and direction of the resultant of two or more non-perpendicular vectors:Break each of the vectors into it’s x- and y- components.It is best to setup a table to nicely organize your components for each vector.
30 Component Table x-component y-component Vector A - (A) Vector B - (B) Vector C - (C)Add more rows if neededResultant - (R)
31 Non-Perpendicular Vectors Once each vector is broken into its x- and y- components :The components along each axis can be added together to find the resultant vector’s components.Rx = Ax + Bx + Cx + …Ry = Ay + By + Cy + …Only then can the Pythagorean Theorem and Tangent function can be used to find the Resultant’s magnitude and direction.
32 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 clown’s total displacement?
33 Practice Problem Picture Step #1: Draw a picture of the problem
34 Practice 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)
35 Step #3: Find the resultant’s components by adding the components along the x- and y-axis. x-componenty-componentVector A - (A)Vector B - (B)Vector C - (C)Resultant - (R)+
36 Step #4: Find the magnitude of the vector by using the Pythagorean theorem. R2 = Δx2 + Δy2
37 Step #5: Find the direction of the vector by using the tangent function. Tan θ = Δy/Δx
38 Practice Problem Answer Step #5: Complete the final answer for the resultant with its magnitude and direction.Practice Problem AnswerResultant displacement = 57.21º