University:
Felician University
Course:
MATH-002 | Basic Math Skills-Algebra
Academic year:

2023
Views:

279

Pages:

7
Author:

StellarSymphony

Vectors Vector: A line segment that has a magnitude and a direction. (LENGTH) Real Life example: A ball flies through the air at a certain speed and in a particular direction. The speed and direction are the velocity of the ball. The velocity is a vector quantity since it has both a magnitude and a direction. Vectors are used to represent things such as force and velocity. We represent vectors by using a directional line segment such as PQ has an initial point P and a Terminal point Q. P----------------->Q The vector v = PQ is the set of all directed line segments of length ||PQ|| which are parallel to PQ. P----------------->Q Two vectors, u and v, are equal if the line segments representing them are parallel and have the same length or magnitude. V---------> U---------> Certainly, here is the text from the image: A vector with initial point (0, 0) is in standard position and is represented uniquely by its terminal point (u₁, u₂). WRITE ⟨u₁, u₂⟩ If v is a vector with initial point P = (p₁, p₂) and terminal point Q = (q₁, q₂), then: The component form of v is v = ⟨q₁ - p₁, q₂ - p₂⟩ The magnitude (or length) of v is ||v|| = √((q₁ - p₁)² + (q₂ - p₂)² Find the component form and magnitude of the vector v with initial point P = (3, -2) and terminal point Q = (-1, 1). FOR COMPONENT FORM: MUST BE TERMINAL - INITIAL! (a) v = ⟨-1 - 3, 1 - (-2)⟩ = ⟨-4, 3⟩ Now using the same vector to find the magnitude. ||v|| = √((-4)² + (3)²) = √(16 + 9) = √25 ||v|| = 5 Given a vector with v = <(2,4), (3,4)>, find the magnitude, ||v||. v = <3-2, 4-4> v = <1, 0> ||v|| = √(1^2 + 0^2) = √1 ||v|| = 1 If the magnitude is equal to one, then this is known as a unit vector. UNIT CIRCLE => RADIUS 1 UNIT UNIT VECTOR => MAGNITUDE 1 UNIT In order to find a unit vector, we simply divide vector v by its magnitude: u = v / ||v|| Find the unit vector of the vector v = <4, 9>: ||v|| = √(4^2 + 9^2) = √(16 + 81) = √97 u = <4, 9> / √97 u = <4/√97, 9/√97> u = <4√97/97, 9√97/97> The standard unit vector v = can be represented as: v = v₁i + v₂j Where v₁ is the horizontal component and v₂ is the vertical component of vector v. The vector sum v₁i + v₂j is called the linear combination of vectors i and j. Let v = <-5, 3>. Write v as a linear combination of the standard unit vectors i and j. (COMPONENT FORM) v = <-5, 3> (LINEAR COMBINATION FORM) v = -5i + 3j