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## Higher Mathematics

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**Higher Mathematics**Unit 3.1 Vectors**A**B 1.Introduction A vector is a quantity with both magnitude and direction. It can be represented using a direct line segment This vector is named or or u**3**5 2 2.Vectors in 3 - Dimensions 2 5 3**-2**4 3**3**2 0**3**-3 -2**P (1, 2)**Q (6, 3) 3 2 1 6 6 - 1 5 3 - 2 1**S (-2, 1)**T (5, 3) 3 1 -2 5 5 - -2 7 3 - 1 2**A (-2, -1)**B (4, 1) 1 - 1 -2 4 4 - -2 6 1- - 1 2**4**4 -3 -3 42 + (-3)2**7**1 C B 1 2 -6 4 A D Add vectors “ Nose-to-tail”**v**u + v u Add vectors “ Nose-to-tail”**A**A -u u B B is the negative of is the negative of**v**-v u + -v u u - v -2 -4 Add the negative of the vector “ Nose-to-tail”**-v**v The Zero Vector Back to the start. Gone nowhere**2v**v 2v has TWICE the MAGNITUDE of v, but v and 2v have the SAME DIRECTION. i.e. They are PARALLEL**P (4, 2)**p The position vector is denoted by If P has coordinates (x , y , z) then the components of the position vector of P are 4 2 The position vector of a point P is the vector from the origin O, to P.**NOT collinear**E D then the vectors are parallel and have a point in common - namely B - , this makes them collinear B C Collinear A**A unit vector is any vector whose length (magnitude) is**one is a unit vector since The vector**-2**4 3 +3 +4 -2 All vectors can be represented using a sum of these unit vectors**The scalar product of the vectors and**is defined as: The scalar product (or “dot” product) is a kind of vector “multiplication”. It is quite different from any kind of multiplication we’ve met before. where q is the angle between the vectors, pointing out from the vertex or**Calculating the angle between two vectors**We have already seen that Rearranging gives And hence we can find the angle between two vectors**3. Perpendicular vectors:**• Provided and are non zero then if then so ie and are perpendiculiar 2. If either or then 4. Some important results using the scalar product • The scalar product is a number not a vector