Math 36 "Fall 08" 7.4 "Vectors" Skills Objectives: * Represent vectors geometrically and algebraically * Find the magnitude and direction of a vector * Add and subtract vectors * Perform scalar multiplication of a vector * Find unit vectors * Express a vector in terms of its horizontal and vertical components Conceptual Objectives: * Understand the di erence between scalars and vectors * Relate the geometric and algebraic representation of vectors Preliminaries: What is the di erence between velocity and speed? A vector quantity is geometrically denoted by a directed line segment, which is a line segment with an arrow representing direction. There are many ways to denote a vector. Geometric Interpretation of Vectors The magnitude of a vector can be denoted in two ways: j~uj or k~uk. We will use the former notation Magnitude: j~uj kthe magnitude of a vector ~u; denoted j~uj ; is the length of the directed line segment.k Page: 1
Note: - Two vectors have if they are parallel and point in the same direction. - Two vectors have if they are parallel and point in opposite direction. Equal Vectors: ~u = ~v Two vectors are equal, ~u = ~v, if they have the same ; and the same : Vector Addition: ~u + ~v Two vectors, ~u and ~v, can be added together using the tail-to-tip rule 1: Translate ~v so that its tail end is located at the tip end of ~u: 2: The sum, ~u + ~v, is the resultant vector from the tail end of ~u to the tip end of ~v Vector Di erence: ~u ~v The di erence, ~u ~v, is the resultant vector from the tip ~v to the tip of ~u Page: 2
Algebraic Interpretation of Vectors Position Vector A vector with its initial point at the origin is called a position vector or in standard position. A position vector, ~u, with its terminal point at (a; b) is denoted: Magnitude: j~uj The magnitude (or norm of a vector), ~u = ha; bi; is: Example 1: (Finding the magnitude of a vector) Find the magnitude of the following vector: a) A = ( 2; 3) and B = (3; 4) Page: 3
b) A = ( 1; 1) and B = (2; 5) Direction Angle of a Vector The positive angle between the x axis and a position vector is called the direction angle, denoted : Example 2: (Finding the direction angle of a vector) Find the direction angle of the following vectors: a) ~u = h3; 4i b) ~u = h 5; 5i Page: 4
Equal Vectors: ~u = ~v The vectors ~u = ha; bi and ~v = hc; di are equal, if Vector Addition: ~u + ~v If ~u = ha; bi and ~v = hc; di; then Vector Di erence: ~u ~v If ~u = ha; bi and ~v = hc; di; then Example 3: (Adding/Subtracting vectors) Let ~u = h1; 2i and ~v = h 5; 4i. Find ~u + ~v Let ~u = h 3; 1i and ~v = h2; 4i: Find ~u ~v Vector Operations Scalar Multiplication: k~u If k is a scalar (real number) and ~u = ha; bi then Scalar multiplication corresponds to - Increasing the length of the vector: - Decreasing the length of the vector: - Changing the direction of the vector: Page: 5
The zero vector, ~0 = h0; 0i; is a vector in any direction with a magnitude equal to zero. Algebraic Properties of Vectors: 1: 6: 2: 7: 3: 8: 4: 9: 5: Example 4: (Using the algebraic properties of vectors) Perform the indicated vector operation, given ~u = h 4; 3i and ~v = h2; 5i a) 5 (~u + ~v) b) 2~u 3~v + 4~u Horizontal and Vertical Components The horizontal component, a, and vertical component, b, of a vector ~u are related to the magnitude of the vector, j~uj, through the sine and cosine functions. Horizontal and Vertical Components The horizontal and vertical components of a vector ~u, with magnitude j~uj and direction angle, are given by The vector, ~u; can then be written as: Page: 6
Example 5: (Finding the horizontal and vertical components of a vector) Find the vector, given its magnitude and direction angle: a) j~uj = 5; = 75 b) j~uj = 8; = 225 Unit Vectors A unit vector is any vector with magnitude equal to one, j~uj = 1: Finding a Unit Vector If ~v is a nonzero vector, then is a unit vector in the same direction as ~v Two important unit vectors are the horizontal and vertical unit vectors i and j: - The unit vector i : - The unit vector j : Example 6: (Finding a unit vector) Find a unit vector in the direction of the given vector: a) ~v = h3; 4i Page: 7
b) ~v = h 4 p 3; 2 p 3i Applications Example 7: A 50 pound weight lies on an inclined bench that makes an angle of 40 with the horizontal. Find the components of the weight directed perpendicular to the bench and also the component of the weight parallel to the inclined bench. Example 8: A force of 500 pounds is needed to pull a speedboat and its trailer up a ramp that has an incline of 16. What is the weight of the boat and its trailer? Page: 8
Example 9: A ship s captain sets a course due west at 12mph. The water is moving at 3mph due north. What is the actual velocity of the ship, and in what direction is it traveling? Example 10: A box weighing 500 pounds is held in place on an inclined plane that has an angle of 10. What force is required to hold it in place? Example 11: A force with a magnitude of 100 pounds and another with a magnitude of 400 pounds are acting on an object. The two forces have an angle of 60 between them. What is the magnitude of the resultant force? Page: 9