![]() The scalar product can also be evaluated from components:.The scalar product is a scalar, it is an ordinary number.The scalar product is also called the ‘dot’ product.a.b = IaI IbI cos(a) where a is the angle between the two vector directions.The scalar product, written a.b can be defined as. ![]() Multiplying by a scalar simply keeps the direction the same, but makes it longer:.Multiplying vectors is more complicated than multiplying simple numbers.effectively you are manipulating all the dimensions simultaneously y x x.This is one of the advantages of working with vectors.The same rules, but with one more component.The same relationships exist between vectors.It doesn’t really matter because all the algebra is the same.There is no way to tell from a vector equation if it is 2-D or 3-D.Vectors in 3-D are an extension of vectors in 2-D.Review of some Mathematics Its usual to use a ˆ above unit vectors, but not always, eg i,j Y (north) 10 5 10 5 X (east) a+b b Two vectors are added by adding the components a= (3,4) b = (6,9) then a + b = (3+6,4+9) Similarly to subtract them a – b = (3-6,4-9) a a-b Y (north) 10 What does it mean to subtract vectors? b -b 5 Subtracting means to go in the opposite direction but for the same distance. Y (north) 10 In fact (5,4) can be made by adding lots of vectors: 5 The sum of lots of ‘short walks’ is the shortest distance between The starting and ending place. Y (north) 10 The vector (5,4) means 5 4 km in the y direction (north) (5,4) total 10 5 X (east) Y (north) 10 The vector (5,4) means 5 5 km in the x direction (east) 10 5 X (east) Y (north) 10 The vector (5,4) means 5 10 5 X (east) ![]() Vectors written as aa a a a & others They all mean the same thing – a direction and magnitude Review of some Mathematics
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