ex) What is the component form of the vector shown in the picture above?
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1 Vectors A ector is a directed line segment, which has both a magnitude (length) and direction. A ector can be created using any two points in the plane, the direction of the ector is usually denoted by the placement of an arrow at the end of the line segment. ex) Pictured here is a ector (named) which has its initial point ( tail point) at ( 2, 1) and its terminal point (the arrow head ) at (2, 3). The component form of a ector is written using the form = ai+ bj. The alue of a is always written as a coefficient of i... which represents the ector s horizontal component. The alue of b is always written as a coefficient of j... which represents the ector s ertical component. ex) What is the component form of the ector shown in the picture aboe? The magnitude of a ector (which is denoted as ) is simply its length. It is calculated by applying the Pythagorean Theorem to the component coefficients a and b. 2 2 For any ector = ai+ bj, its magnitude is = a + b ex) What is the magnitude of the ector from the picture aboe? (Use the component form)
2 ex) Find the component form and the magnitude of the ector with initial point at ( 3, 11) and terminal point at(9, 40). (Approximate the magnitude to 2 decimal places.) The two main operations with ectors are ector addition and scalar multiplication. These operations can be done algebraically and graphically. Ex) For the ectors u= i+ 3j and = 2i j Plot ectors with their initial points at the origin and determine the following ector combinations. (a) u+ Find the sum algebraically (using component forms) Find the sum graphically (using parallelogram law) Calculate u+
3 (b) 2u+ 3 Find the sum algebraically (using component forms) Find the sum graphically (using parallelogram law) The 2 applied to u and the number 3 applied to are examples of scalar multiplication. The scalars will scale the length of each ector making them longer. Calculate 2u+ 3 (c) u Find the sum algebraically (using component forms) Find the sum graphically (using parallelogram law) Calculate u
4 Direction Angle for a Vector The direction angle is always considered to be the standard position angle starting on the positie x axis rotating counterclockwise to the ector s position. It can be determined by the formula: horizontal component tanθ = ertical component 1 To get the angle you ll need to use TAN on your calculator... BUT MAKE SURE YOUR ANGLE IS LOCATED IN THE CORRECT QUADRANT! ex) Determine the direction angle for the ector = 5i+ j. (IT ALWAYS HELPS TO SKETCH THE VECTOR FIRST) ex) Determine the direction angle for the ector = 4i+ 7j. 1 (SKETCH THE VECTOR FIRST AND BE CAREFUL USING TAN )
5 Unit Vectors When you want to presere the direction of a certain ector but apply a different length to it, you ll need to transform that ector in to a unit ector... essentially a ector of length 1. To get a unit ector, you diide the components by the magnitude: unit ector = ex) What is the unit ector which has the same direction as = 4i 3j? ex) A force of 1200 lbs is applied in the direction of the ector = 4i 3j. What are the components of this force ector? (Call the force ector F)
6 Decomposing a Vector When you already know the magnitude and direction angle, θ, of a ector, you can write it in component form using the formulas Horizontal Component cosθ Vertical Component sinθ Creating the ector = ( cos θ) i+ ( sin θ) j ex) Find the horizontal and ertical components of the ector with a length of = 800 and a direction ofθ = 145. (Round components to 2 decimal places). ex) A jet is flying in a direction of N 20 E with a speed of 500 mi/h. Represent the elocity of this jet as a ector in component form. (2 decimal place rounding)
7 Resultant Vectors The resultant of two or more ectors is the result of all of the ectors acting on the same object at the same time. A resultant is simply their ector sum. ex) Two tugboats are pulling a barge due north through a channel. One tugboat is pulling with a force of 3500 lbs at a heading of N 20 E and the other tugboat is pulling with 4000 lbs of force at a heading of N 25 W. (See the diagram.) a) What are the component forms of the force ector for each tugboat? b) Calculate the resultant ector. This is the ector sum of adding the tugboat ectors together. c) What is the magnitude and the direction of the resultant ector? Gie the direction as a bearing.
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