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The definition of the product of two vectors is: v w a1a2 b1b2 where v a1i b1 j and w a2i b2 j If v 2i 5 j and w 4i 1 j, find v w v w 2 4 5 1 8 5 3 This is called the dot product. Notice the answer is just a number NOT a vector. The dot product is useful for several things. One of the important uses is in a formula for finding the angle between two vectors that have the same initial point. If u and v are two nonzero vectors, the angle , 0 < , between u and v is determined by the formula cos u v u v v u Technically there are two angles between these vectors, one going the "shortest" way and one going around the other way. We are talking about the smaller of the two. Find the angle between u = 2i j and v = 4i + 3j. uv cos u v u v 2 4 13 8 3 5 u 2 1 2 2 5 v 4 3 16 9 25 5 2 v 4i 3j u 2i j 2 uv cos u v cos 1 5 5 5 1 5 1 63.4 5 Find the angle between the vectors v = 3i + 2j and w = 6i + 4j vw 18 8 cos v w 13 52 cos 1 0 1 w 6i 4 j v 3i 2 j 26 676 1 What does it mean when the angle between the vectors is 0? The vectors have the same direction. We say they are parallel because remember vectors can be moved around as long as you don't change magnitude or direction. If the angle between 2 vectors is , what would their dot w = 2i + 8j 2 product be? uv cos u v 2 Since cos 2 is 0, the dot product must be 0. Vectors u and v in thisvcase are- called orthogonal. = 4i j (similar to perpendicular but refers to vectors). Determine whether the vectors v = 4i - j and w = 2i + 8j are orthogonal. compute their dot product and see if it is 0 v w 4 2 18 0 v w 0 The vectors v and w are orthogonal. A use of the dot product is found in the formula below: The work W done by a constant force F in moving an object from A to B is defined as W F AB This means the force is in some direction given by the vector F but the line of motion of the object is along a vector from A to B Find the work done by a force of 50 kilograms acting in the direction 3i + j in moving an object 20 metres from (0, 0) to (20, 0). Let's find a unit vector in the direction 3i + j Remember to get a unit vector, divide a vector by it's magnitude 3i + j W F AB 20i + 0j (20, 0) 3 1 W 50 i j 20 i 0 j 10 10 3i j 3 1 2 2 10 3 1 u i j 10 10 Our force vector is in this direction but has a magnitude of 50 so we'll multiply our unit vector by 50. 3000 W 948.68 m kgs F 50 10 3 1 i j 10 10 Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au