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9.4 Vectors in the Plane vector: (v or v )a mathematical object with both magnitude and direction. Vectors are depicted in two ways: component form: the form <a,b> used to denote a vector. standard representation: the arrow from the origin to the point (a, b) 9.4 Vectors in the Plane magnitude of v: denoted |v|, it is the length of the vector. direction of v: the direction in which the arrow points initial point: the starting point of the arrow, also called the “tail” terminal point: the ending point of the arrow, also called the “head” Head Minus Tail Rule If an arrow has initial point x1 , y1 and terminal point x2 , y2 it represents the vector x2 x1, y2 y1 equivalent vectors: two arrows which represent the same vector. Example 1: Show that the arrows from (2, 3) to (4, 7) and from (-1, 1) to (1, 5) are equivalent x2 x1 , y2 y1 x2 x1 , y2 y1 4 2,7 3 1 1,5 1 2,4 2,4 Magnitude If v is represented by the arrow from x1 , y1 to x2 , y2 then v x x y y 2 2 If v = <a, b> then 1 2 2 1 v a 2 b2 Example 2: Find the magnitude of the vector from the point (-2, 3) to (4, 6) v x2 x1 2 y2 y1 2 v 4 22 6 32 v 62 32 v 36 9 v 45 v 3 5 Vector Addition and Scalar Multiplication Let u=<u1, u2> and v=<v1, v2> and let k be a real number (scalar). Then the sum of the vectors is: u v u1 v1 , u2 v2 The product of the scalar k and a vector is: ku k u1 , u2 ku1 , ku2 Example 3: Let u=<3, 2> and let v=<-1, 3>. Find 3u + 2v 3u 2v 3 3,2 2 1,3 9,6 2,6 7,12 Unit Vectors unit vector: a vector with magnitude 1 unit vector in the direction of v: to find the unit vector u in the direction of v use the following formula: v 1 u v v v Unit Vectors standard unit vectors: the vectors i=<1, 0> and j=<0, 1> are unit vectors in the direction of the positive x-axis and y-axis respectively. linear combination: the expression v=ai+bj used to represent the vector <a, b> Example 4: Find a unit vector in the direction of the vector 2 2 v=<3, 4> v a b 2 2 v 3 4 v u v v 9 16 u 3,4 5 3 4 u , 5 5 v 25 v 5 Direction Angles If v has the direction angle θ as measured from the x-axis, then the component form of the vector can be found by the following formula: v v cos , v sin Example 5: A basketball is shot at a 60° angle with an initial speed of 10 m/s. Find the component form of this vector. v v cos , v sin v 10 cos 60,10 sin 60 v 5,5 3 9.4 HW Assignment Pg. 603: #s 18-52 evens, 58-62 evens