chapter 1 physics

# chapter 1 physics - Geometric vs Componentwise Vector...

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Geometric vs Componentwise Vector Addition Learning Goal: To understand that adding vectors by using geometry and by using components gives the same result, and that manipulating vectors with components is much easier. Vectors may be manipulated either geometrically or using components. In this problem we consider the addition of two vectors using both of these two methods. The vectors and have lengths and , respectively, and makes an angle from the direction of . Vector addition using geometry Vector addition using geometry is accomplished by putting the tail of one vector (in this case ) on the tip of the other ( ) and using the laws of plane geometry to find the length , and angle , of the resultant (or sum) vector, : 1. , 2. Vector addition using components Vector addition using components requires the choice of a coordinate system. In this problem, the x axis is chosen

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Geometric vs Componentwise Vector Addition along the direction of . Then the x and y components of are and respectively. This means that the x and y components of are given by 3. , 4. . Part A Which of the following sets of conditions, if true, would show that the expressions 1 and 2 above define the same vector as expressions 3 and 4? Check all that apply. ANSWER: The two pairs of expressions give the same length and direction for . The two pairs of expressions give the same length
Geometric vs Componentwise Vector Addition and x component for . The two pairs of expressions give the same direction and x component for . The two pairs of expressions give the same length and y component for . The two pairs of expressions give the same direction and y component for . The two pairs of expressions give the same x and y components for . Correct To show that the two pairs of expressions (for and and for and ) define the same vector you can show that any of the sets of conditions listed above are met except They give the same length and x component for . They give the same length and y component for . If you consider just the first set of conditions, showing that the two sets of expressions have the same length and x component will imply that the y component has the correct magnitude. However, there is no way to know the sign (i.e., direction) of the y component. To show that the pairs of expressions given above define the same vector , we would need to show that they give the same length and the same x and y components. We will do this in the questions that follow. Part B We begin by investigating whether the lengths are the same.

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Geometric vs Componentwise Vector Addition Find the length of the vector starting from the components given in Equations 3 and 4. Hint B.1 Apply the Pythagorean theorem Hint not displayed Express in terms of , , and . ANSWER:
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## This note was uploaded on 06/01/2010 for the course PHY 2048 taught by Professor Yifu zhu during the Summer '09 term at FIU.

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chapter 1 physics - Geometric vs Componentwise Vector...

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