Unit 7 Vectors

University:
University of California, Berkeley
Course:
MATH W54 | Linear Algebra and Differential Equations
Academic year:

2021
Views:

208

Pages:

3
Author:

basteishx23y

Unit 7 Vectors ● Vectors have both direction and magnitude. ○ The length of the ray indicates the magnitude. ○ The arrow on the ray indicates the direction. ● Vectors can be added: ○ Vector addition is commutative and distributive. ● Vectors can be multiplied by a scalar(dilation). ● Vectors can be defined by an initial and terminal point on a coordinate plane. ○ 𝑣 = 𝑎𝑖 + 𝑏𝑗 ● Magnitudes of Vectors ○ | |𝑣| | = 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 # ○ Always nonnegative: | |𝑣| |=| |𝑣| |, | |𝑣| |> 0 ● Unit Vector ○ 𝑢=| 𝑣 𝑣𝑒𝑐𝑡𝑜𝑟 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 = |𝑣| | 𝑣𝑒𝑐𝑡𝑜𝑟 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 ● Vectors are written with respect to angle measure. ○ 𝑣 = | |𝑣| |(𝑐𝑜𝑠𝑎𝑖 + 𝑠𝑖𝑛𝑎𝑗) ● Static Equilibrium: forces that can be represented by vectors. ○ If the object is at rest, and the sum of all forces acting on the object is zero, then, an object is in static equilibrium. The Dot Product ● If 𝑣 = 𝑎1 𝑖 + 𝑏1 𝑗 𝑎𝑛𝑑 𝑤 = 𝑎2 𝑖 + 𝑏2 𝑗, then… ○ 𝑣 ⋅ 𝑤 = 𝑎1 𝑎2 + 𝑏1 𝑏2 ○ For dot product, the commutative and associative properties can be applied. ● Angles between vectors: ○ If u and v are nonzero vectors, the angle θ, 0 ≤ 𝜃 ≤ 𝜋, between the two vectors is determined by: ■ 𝑐𝑜𝑠𝜃 = | 𝑢⋅𝑣 |𝑢| |||𝑣|| ● Orthogonal/Perpendicular vectors occur when the angle between two nonzero vectors is 𝜋 2 … i.e. 𝑣 ⋅ 𝑤 = 𝑜 ● Applications: work done by a constant force. ○ Work = (magnitude of Force vector in direction of movement)(distance) ○ Units: foot-pounds or newton-meters(Joules) Parametric Equations ● Let 𝑥 = 𝑓(𝑡) 𝑎𝑛𝑑 𝑦 = 𝑔(𝑡), where 𝑓 𝑎𝑛𝑑 𝑔are two functions whose common domain is some interval I. The collection of points defined by (𝑥, 𝑦) = (𝑓(𝑡), 𝑔(𝑡))is called a parametric point. ● The equations 𝑥 = 𝑓(𝑡) 𝑎𝑛𝑑 𝑦 = 𝑔(𝑡) are known as parametric equations. ● T is called the parameter(time). ● Projectile motion ○ The location of an object as it travels through space depends on: ■ Horizontal motion: Velocity, angle,time ● 𝑥 = 𝑣0 𝑡𝑐𝑜𝑠𝜃 ○ v0=initial velocity ■ Vertical motion: velocity, angle, gravity, time 1 ● 𝑦 = − 2 𝑔𝑡 2 + 𝑣0 𝑡𝑠𝑖𝑛𝜃+h ○ h = initial height ○ g = gravity = 9.8 m/s2 Polar Graphs