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UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS MATH 1131 MATHEMATICS 1A ALGEBRA. Section 2: - Vector Geometry. In Chapter 1, see looked at the rudiments of vector geometry. We now have enough machin- ery to study this subject in greater depth. Distances and Lengths: We saw earlier that: in higher dimensions, we can define the length of a vector as fol- lows: Definition: A vector x = a 1 a 2 . . . a n in R n has length | x | given by | x | = p a 2 1 + a 2 2 + · · · + a 2 n . The distance between two points A and B in R n will be defined as the length of the vec- tor -→ AB , in other words, if A has position vector a = a 1 a 2 . . . a n and B has position vector b = b 1 b 2 . . . b n , then the length of -→ AB is | -→ AB | = | b - a | = p ( b 1 - a 1 ) 2 + · · · + ( b n - a n ) 2 . A vector which has unit length is called a unit vector . Any non-zero vector can be made into a unit vector by dividing by its length. Dot Product: We define the dot product of two vectors a = a 1 a 2 a 3 and b = b 1 b 2 b 3 by a · b = a 1 b 1 + a 2 b 2 + a 3 b 3 . Note that this is a scalar, and so the dot product is often called a scalar product . Ex: Find the dot product of a = 1 2 3 and b = - 1 7 - 3 1
The dot product also has a geometric interpretation in R 2 and R 3 . Indeed for non-parallel vectors a and b in R 3 , we consider the following triangle. c b a We can write c = a - b and so taking lengths, we have | c | 2 = ( a 1 - b 1 ) 2 + ( a 2 - b 2 ) 2 + ( a 3 - b 3 ) 2 = | a | 2 + | b | 2 - 2( a 1 b 1 + a 2 b 2 + a 3 b 3 ) = | a | 2 + | b | 2 - 2 a · b . Also, if we write down the cosine rule for this triangle, we have | c | 2 = | a | 2 + | b | 2 - 2 | a || b | cos θ , where θ is the angle between the vectors a and b . Comparing the two expressions we see that a 1 b 1 + a 2 b 2 + a 3 b 3 = a · b = | a || b | cos θ. This formula is fundamental and should be committed to memory. Notice that if the angle between a and b is 90 we have a · b = a 1 b 1 + a 2 b 2 + a 3 b 3 = 0 . Ex: Find the dot product and hence the acute angle between the vectors a and b if a = 1 - 1 2 and b = 2 1 1 . For vectors in R n we can define the dot product directly as follows. 2
Definition: The dot product of two vectors a and b in R n is given by a · b = a 1 b 1 + · · · + a n b n . The dot product has the following properties: 1. a · a = | a | 2 and so | a | = a · a . 2. a · b is a scalar (which is why the dot product is sometimes called the scalar product ). 3. a · b = b · a . (Commutative law). 4. a · ( λ b ) = λ ( a · b ). 5. a · ( b + c ) = a · b + a · c . (Distributive law). The proofs of these are easy. Ex: Prove that the diagonals of a rhombus are perpendicular. In higher dimensions we can define the angle θ between two vectors a and b by cos θ = a · b | a || b | . Orthogonality: Two vectors a and b are said to be orthogonal if a · b = 0. 3
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