VECTORS.

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Presentation transcript:

VECTORS

Scalar Multiplication – Resizing Vectors Any vector can be resized by multiplying it by a real number (scalar). Multiplying by positive scalar changes magnitude only. Multiplying by a negative scalar changes the magnitude and its direction.

Resizing Written Vectors Example: Given u = <3, 5>, find 4u. “distribute” 4u = 4<3, 5> = <4x3, 4x5> = <12, 20>

Adding Vectors – Geometrically “Parallelogram Method” Given two vectors, to add them geometrically, you can use a parallelogram. First, join the vectors initial points (tails). Second, create two more vectors that are equal to the original vectors. Place them where the tails meet the heads of the first set and join their heads to make a parallelogram. Finally, the resultant vector of this addition is the diagonal from the joined tails to the joined heads.

Adding Vectors – Geometrically “Parallelogram Method” Join the tails of the two vectors you are adding. Create two equal vectors. Join the new vectors’ tails to the heads of the original. Draw the diagonal from the TAIL to the HEAD. u u + v v

Adding Vectors in Written Form Adding vectors in written forms is fairly simple. Basically you just have to follow the order of operations. In component form: Multiply through by any scalars. Add horizontal components, Add vertical components In Linear combinations: Combine like terms.

Adding Vectors in Written Form Examples. Given u = <3, 5> and v = <2, -4> find the following vectors. 3u + 2v = 2u - v = 3<3, 5> + 2 <2, -4> = <9, 15> + <4, -8> = <13, 7> 2<3, 5> - <2, -4> = <6, 10> + <-2, 4> = <4, 14>

Unit Vectors A unit vector is a vector of magnitude 1 (in any direction). To find a unit vector in a specific direction (the direction of another given vector), you must “divide” the given vector using scalar multiplication so that the new vector’s magnitude is 1. Find the magnitude of the given directional vector. Multiply by the reciprocal of the magnitude.

Unit Vectors Example: Find the unit vector in the same direction as <-3, 4>. 1. Find the magnitude of <-3, 4> ||<-3, 4>|| = 3 2 + 4 2 = 9+16 = 25 =5 2. Find the unit vector by multiplying by the reciprocal. 1 5 <-3, 4> = <− 3 5 , 4 5 >

Assignment Assignment 2 Alternate Text – On Blog p. 433-434 #13-22, 25-28, 35-40, 45-46