Unit 2: Algebraic Vectors

Schedule for Algebraic Vectors Today: Intro and Notation Tomorrow: Operations on algebraic vectors (adding & scalar multiplicaiton) Friday: The Dot Product ( 𝑎 ∙ 𝑏 = 𝑎 𝑏 cos𝜃) Monday: The Cross Product (multiplying 2 vectors - 𝑎 × 𝑏 ) Tuesday: Applications of the Dot Product and Cross Product Wednesday: Review Thursday: Test Friday: Warm-up to Unit 3: Linear Systems

2.1 Introduction to Algebraic Vectors Position Vector: The vector 𝑂𝑃 with a tail at the origin, O and the tip at the point P.

Example 1: The geometric vector 𝑣 is defined to have a magnitude of 10 and a direction of N 45 ° E. Determine the components of the position vector.

Example 2: Draw vector 𝑂𝑃 =[−2,3] and determine its magnitude & direction. ∴ If you have a vector 𝑢 =[𝑎,𝑏] then 𝑢 = 𝑎 2 + 𝑏 2 and 𝜃= tan −1 𝑏 𝑎 .

Unit Vectors We can express any vectors in the 𝑥𝑦-plane as a sum of scalar multiples of the vectors 𝑖 and 𝑗 , where 𝑖 =[1,0] and 𝑗 = 0,1 . Example: 𝑂𝑃 = 5,6 =5 𝑖 +6 𝑗 In general, ordered pair notation and unit vector notation are equivalent.

3 Dimensions (𝑅 3 )

3 Dimensions (𝑅 3 ) We can express any vectors in the 𝑥𝑦𝑧-plane as a sum of scalar multiples of the vectors 𝑖 , 𝑗 , 𝑘 where 𝑖 = 1,0,0 , 𝑗 = 0,1,0 and 𝑘 = 0,0,1 . Ordered triple: (𝑥,𝑦,𝑧) Example: 𝑂𝑃 = 3,5,4 =3 𝑖 +5 𝑗 +4 𝑘

3 Dimensions (𝑅 3 ) If you have a vector 𝑢 =[𝑎,𝑏,𝑐] then 𝑢 = 𝑎 2 + 𝑏 2 + 𝑐 2 and the direction angles are 𝛼,𝛽, 𝛾, where 0≤𝛼,𝛽, 𝛾≤ 180 ° such that: 𝑐𝑜𝑠𝛼= 𝑎 𝑂𝑃 𝑐𝑜𝑠𝛽= 𝑏 𝑂𝑃 𝑐𝑜𝑠𝛾= 𝑐 𝑂𝑃

Example: If you have a vector 𝑢 = −2,3,0 , find 𝑢 and the direction angles.