Lecture Notes (Vectors)

Transcription

1 Lecture Notes (Vectors) Intro: - up to this point we have learned that physical quantities can be categorized as either scalars or vectors - a vector is a physical quantity that requires the specification of both direction and magnitude (size); vectors are represented italic type with an arrow over the letter or by boldface italic type - a scalar is a quantity that can be completely specified by its magnitude with appropriate units; scalars are represented by italic type - while scalars can be manipulated using the rules of ordinary arithmetic, vectors require special procedures; this is what we will be discussing today Properties of Vectors: - vectors may be represented two different ways; they may be represented algebraically or graphically - an example of algebraic representation might be v 50 m/s at 30 - a graphical representation would be an arrow or an arrow tipped line segment Ex. 50 m, 30

2 - the arrows are drawn to scale so that its length represents the magnitude of the vector and the arrow points in the specified direction of the vector - vectors, such as displacement, dependd only on the initial and final positions, so the vector is independentt of the path taken between these two points Equal Vectors: - two vectors A and B are equal if they have the same magnitude and same direction; this allows us to translate a vector parallell to itself in a diagram without affecting the vector

3 - you should note that vectors are not equal if the have the same length but do not point in the samee direction Adding Vectors: - when two or more vectors are added, they must all have the same units; Ex. you cannot add a velocity vector to a displacement vector; scalars follow this rule as well, it would be meaningless to add temperature and volumes - to add vectors graphically, draw the first vector, A, on a piece of graph paper to some scale such as 1 cm = 1 m; draw A so that its direction is specified relative to a coordinate system - then draw vector B to the same scale and with the tail of B starting at the tip of A; vector B must be drawn along the direction that makes the proper angle relative to vector A - the resultant vector (R = A + B) is thee vector drawn from the tail of A to the tip of B; this is called the triangle method of addition

4 - for example, if you walked 3.0 m toward the east and then 4.0 m toward the north, as shown below,, you would find yourself 5.0 m from where you started measured at an angle of 53 north of east - your total displacem ment is the vector sum of the individual displacements - if the two vectors to be addedd together are at a right angle to one another, you can find the magnitude by using the Pythagorean Theorem; a 2 + b 2 = c 2 - if the two vectors to be addedd are not at a rightt angle, you can find the magnitude by using the Law of Cosines; R 2 = A 2 + B 2-2AB(cos Θ) Vector Subtraction: - the subtraction of vectors is similar to vector addition, but uses the addition of a negativee vector - we define A - B as vector -BB added to vector A A - B = A + (-B)

5 Vector Direction: - the direction of a vector is defined as the angle that the makes with the x-axis measured counter-clockwise vector Vector Components: - consider a vector in a particular plane; it can be expressed as a sum of two other vectors called the components of the original vector - the components are chosen along two perpendicular directions - the processs of finding these perpendic cular components is knownn as vector resolution - the components of a vector can be found by drawing lines out from the tip of the vector making them perpendicular to the x and y axes - if the original vector was v, then we would name its components v x and v v y

6 Algebraic Addition of Vectors: - you can algebraically add vectors by first breaking up the vector into its components - next, you then add the x-components the resultant to find the x-component of - similarly, you can find the resultant y-component by adding the y-components - if you have a right triangle, you can find the magnitude resultant vector by using the Pythagorean theorem of the - if you do not have a right triangle you may use the Law of Cosiness if you have two side lengths and an included angle - you may use the trigonometric functions (sine,, cosine, and tangent) only if you have a right triangle

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