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ps01_solns - Physics 1311 Spring 2010 PS#1 Solutions Page 1...

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Physics 1311 Spring 2010 PS #1 Solutions Page 1 of 6 Physics 1311 Problem Set #1 Solutions Problem 1: Determine whether each of the following statements is true or false: (a) The magnitude of the sum of two vectors is always greater than the magnitude of either vector. (b) The magnitude of the sum of two vectors is always less than the sum of the magnitudes of the two vectors. (c) A vector’s component can never be larger than the magnitude of the vector. (d) It is possible for a component of a vector to be zero if the vector itself is not zero. (e) It is possible for a vector to be zero, while a component of the vector is not zero. (a) False. For an extreme example, A + ( - A ) = 0; the sum is zero, but A can have an arbitrarily large magnitude. (b) False. If the two vectors point in the same direction, then the magnitude of the sum | A + B | could be equal to the sum of the magnitudes | A | + | B | . (c) True. The magnitude of a vector can be found from its components using the Pytha- gorean theorem: | A | = A 2 x + A 2 y . This shows that the magnitude has to be at least as large as any of the components. (d) True. For example, a vector oriented horizontally has no vertical component. (e) False. The squared magnitude of a vector is the sum of the squares of its components, according to the Pythagorean theorem. If one of the components is nonzero, there’s no way for the perpendicular component to “cancel it” so that the resultant is zero. Said another way, a zero vector has to have all zero components. Problem 2: Which of the statements about the vectors depicted by these three arrows are correct? t r s By using the graphical “tip to tail” rule for vector addition, it can be seen that the two angled vectors add together to produce the horizontal vector: r + s = t . There are a couple of other ways to write this, by rearranging the three terms: r = t - s s = t - r .
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Physics 1311 Spring 2010 PS #1 Solutions Page 2 of 6 Problem 3: The vector g = 2 , - 4 , 5 , and the scalar h = - 5 . What is g + h ? A vector can be multiplied by a scalar. However, vectors can’t be added to scalars; they can only be added to other vectors. (Also, the vectors being added must have the same dimensions, otherwise the sum is physically meaningless.) Problem 4: The two vectors a and b in the figure have equal magnitude of 31 m and the angles are θ 1 = 21 and θ 2 = 96 . (a) Find the x-component of their vector sum r . (b) Find the y-component of their vector sum r . (c) What is the magnitude of their vector sum
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