Vectors Vectors are represented by a directed line segment its length representing the magnitude and an arrow indicating the direction A B or u u This.

What are Rays, Lines and Line Segments? | Geometry | Don’t Memorise
What are Rays, Lines and Line Segments? | Geometry | Don’t Memorise

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Vectors Vectors are represented by a directed line segment its length representing the magnitude and an arrow indicating the direction A B or u u This vector is named

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v === -v = w == = -w-w = These are also known as COLUMN VECTORS C D v E F w

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C D v E F w v == This is also known as the components of v

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Q P a b It can be calculated using Pythagoras ExampleIf

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Vectors are only equal if they have the same magnitude and direction. Equal Vectors a b c d For vectors and

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For vectors and Addition Of Vectors

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== i) Find the components of = =

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The Zero Vector is called the zero vector written 0 If find the components of

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Subtraction of Vectors Page 236 Exercise 13D questions 1 to 3

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Multiplication by a scalar

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Unit Vectors For any vector v there exists a parallel vector u of magnitude 1 unit. This is called a Unit Vector. i.e. Find the components of the unit vector u parallel to vector Since the magnitude of v is 5, the unit vector u must be 1 / 5 v

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Position Vectors If P and Q have coordinates (4,8) and (2,3) find the components of

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Collinearity We have seen that if a vector v = ku then v must be parallel to u. If vectors v and u also have a point in common then because they are parallel they must lie on the same line so by definition must be collinear.

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Prove that the points A(2,4), B(8,6) and C(11,7) are collinear. B is a point in common to both AB and BC so A, B and C are collinear

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Section Formula If p is the position vector of the point P that divides AB in the ratio m:n then: A B P m n

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A and B have coordinates (3,2) and (7,14) respectively. Find the coordinates of the point P that divides AB in the ratio 1:3 1. Draw a quick sketch (3,2) (7,14) P 1 3

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3 Dimensional Vectors x y z A The point A has a position relative to the x y and z axis 3 4 6 A(3, 4, 6)

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Find the coordinates of P y x z P 4 2 1 P(4, 2, 1) O

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Find the coordinates of Q y x z Q -2 -3 Q(-1, -2, -3)

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3D Unit Vectors A vector can also be defined in terms of i, j and k where i, j and k are unit vectors in the x, y, and z directions respectively. y x z i 1 j 1 1k1k In component form the vectors are written as

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Any vector can be expressed as a combination of its components.

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Properties of 3D vectors P Q 4 2 5 R S

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Addition / Subtraction Scalar Position Vector

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Section Formula A (4, -6, 12) B (4, 4, -3) P 3 2

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The Scalar Product For two vectors a and b, the scalar product is defined by Where is the angle between a and b, The scalar product is also known as the dot product. The vectors must be directed away from the point of intersection.

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If a and b are perpendicular then a. b = 0

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Component Form of a Scalar Product

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Angle between vectors

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Other Vector Facts PROOF

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30 0 p r q

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