Vectors Geometry KUS objectives BAT solve problems with

Vectors: Geometry • KUS objectives BAT solve problems with vector geometry; understand the basics of vector geometry including addition and scalar multiplication Starter: name four types of transformation of a shape Reflection over a given line Rotation cw or acw around a given point by an angle Translation by a vector Enlargement by a scale factor from a given point

Notes 1 Direction and Magnitude Q A scalar quantity has only a magnitude (size) A vector quantity has both a magnitude and a direction Scalar Vector The distance from P to Q is 100 m Scalar A ship is sailing at 12 km/h From P to Q you go 100 m north P N Vector 60° A ship is sailing at 12 km/h on a bearing of 060°

Notes 2 Q Equal vectors have the same magnitude and direction S P A Common way of showing vectors is using the letters with an arrow above Alternatively, single letters can be used… R PQ = RS a b

North Notes 3 7 p 4 Use arctan The modulus value of a vector is another name for its magnitude a Eg) The modulus of the Vector a is |a| The modulus of the vector PQ is |PQ| Example The vector a is directed due east and |a| = 12. Vector b is directed due south and |b| = 5. Find |a + b| a + b b Use Pythagoras’ Theorem

Q Notes 4 a Adding the vectors PQ and QP gives a Vector result of 0 Q P Vectors of the same size but in opposite directions have opposite signs (eg) + or - -a P B C A a b b a+b a

WB 1 Q In the diagram opposite, find the following vectors in terms of a, b, c and d a c P -c + a -b + a Or Or c - a S a - b -a + b + d Or b + d - a -d - b + c Or c – b - d b R d T

WB 2 parallel vectors Any vector parallel to a may be written as λa, where λ (lamda) is a non-zero scalar (ie - represents a number…) a) Show that the vectors 6 a + 8 b and 9 a + 12 b are parallel… The second Vector is a multiple of the first, so they are parallel b) Show that the vectors 5 b - 2 a and 14 a - 35 b are parallel…

Triangle Law for vectors Two vectors can be added using the ‘Triangle Law’ b a It is important to note that vector a + b is the single line from the start of a to the end of b. a + b Vector a + b is NOT the two separate lines! Example Draw a diagram to show the vector a + b + c a b c

WB 4 Triangle Law for vectors ii) Careful with diagrams ! Use the sine rule

WB 5 In the diagram, PQ = 3 a, QR = b, P SR = 4 a and PX = k. PR. Find in terms of a, b and k: S = k. PR = k(3 a + b) = 4 a - b = -b + a + k(3 a + b) = (3 k + 1)a + (k – 1)b e) Use the fact that X lies on SQ to find the value of k Q X = 3 a + b – 4 a = b – a 3 a = (3 k + 1)a + (k – 1)b 4 a b R

Equate the coefficients of a and b So D is the midpoint of both MN and ST So MN and ST bisect each other

Effectively, if the two vectors are equal then the coefficients of a and b must also be equal

• KUS objectives BAT solve problems with vector geometry; understand the basics of vector geometry including addition and scalar multiplication self-assess One thing learned is – One thing to improve is –

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