3 Vectors. 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan
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1 Chapter 3 Vectors
2 3 Vectors 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 2
3 3 3-2 Vectors and Scalars Physics deals with many quantities that have both size and direction. It needs a special mathematical language-the language of vectors-to describe those quantities. In this chapter: we will focus on the basic language of vectors.
4 4 3-2 Vectors and Scalars The simplest example is a displacement vector. If a particle changes position from A to B, we represent this by a vector arrow pointing from A to B. Arrows are used to represent vectors. The length of the arrow signifies magnitude The head of the arrow signifies direction Sometimes the vectors are represented by bold lettering, such as vector a. Sometimes they are represented with arrows on the top, such as A
5 5 3-3 Adding vectors geometrically The vector sum, or resultant Is the result of performing vector addition Represents the net displacement of two or more displacement vectors Vector a and vector b can be added geometrically to yield the resultant vector sum, s. Place the second vector, b, with its tail touching the head of the first vector, a. The vector sum, s, is the vector joining the tail of a to the head of b.
6 6 3-3 Adding vectors geometrically; Some rules Vector addition is commutative We can add vectors in any order Vector addition is associative We can group vector addition however we like
7 7 3-3 Adding vectors geometrically; Some rules A negative sign reverses vector direction We use this to define vector subtraction Only vectors of the same kind can be added (distance) + (distance) makes sense (distance) + (velocity) does not
8 3-3 Adding vectors geometrically Example: 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 8
9 9 3-4 Components of vectors The component of a vector along an axis is the projection of the vector onto that axis. The process of finding the components of a vector is called resolution of the vector. In 3-dimensions, there are three components of a vector along predefined x-, y-, and z-axes.
10 Components of vectors The components of a vector can be positive or negative. We find the components of a vector by using the right triangle rules. Where θ is the angle the vector makes with the positive x axis, and a is the vector length ( a ). They are unchanged if the vector is shifted in any direction (but not rotated).
11 Components of vectors Angles may be measured in degrees or radians Recall that a full circle is 360, or 2π rad Know the three basic trigonometric functions
12 Example 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 12
13 3-4 Components of vectors 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 13
14 Unit Vectors A unit vector Has magnitude 1 Has a particular direction Lacks both dimension and unit Is labeled with a hat: ^ Unit vectors pointing in the x-, y-, and z-axes are usually designated by ˆ respectively iˆ, ˆ, j k Therefore vector, a, with components a x and a y in the x- and y-directions, can be written in terms of the following vector sum: We use a right-handed coordinate system Remains right-handed when rotated
15 3-6 Adding vectors by components 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 15 If then Therefore, two vectors must be equal if their corresponding components are equal. The procedure of adding vectors also applies to vector subtraction. Therefore, where
16 3-6 Adding vectors by components Example: 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 16
17 Adding vectors by components Example (Continued):
18 Vectors and the Laws of Physics Freedom of choosing a coordinate system Relations among vectors do not depend on the origin or the orientation of the axes. Relations in physics are also independent of the choice of the coordinate system. We can rotate the coordinate system, without rotating the vector, and the vector remains the same All such coordinate systems are equally valid
19 Multiplying Vectors A. Multiplying a vector by a scalar Multiplying a vector by a scalar changes the magnitude but not the direction: B. Multiplying a vector by a vector: Scalar (Dot) Product The scalar product between two vectors is written as: It is defined as: Here, a and b are the magnitudes of vectors a and b respectively, and f is the angle between the two vectors. The right hand side is a scalar quantity.
20 Multiplying Vectors C. Multiplying a vector with a vector: Vector (Cross) Product The vector product between two vectors a and b can be written as: The result is a new vector c, which is: Here a and b are the magnitudes of vectors a and b respectively, and f is the smaller of the two angles between a and b vectors. The right-hand rule allows us to find the direction of vector c.
21 Multiplying Vectors C. Multiplying a vector with a vector: Vector (Cross) Product The upper shows vector a cross vector b, the lower shows vector b cross vector a.
22 Multiplying Vectors C. Multiplying a vector with a vector: Vector (Cross) Product
23 Multiplying Vectors The cross product is not commutative To evaluate, we distribute over components: Therefore, by expanding
24 3-8 Multiplying Vectors Example: 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 24
25 3-8 Multiplying Vectors Example : 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 25
26 Example: 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 26
27 27 3 Solved Problems 1. Three vectors a, b, and c each have a magnitude of 50 m and lie in an xy-plane. Their directions relative to the positive direction of the axis are 30, 195, and 315, respectively. What are (a) the magnitude and (b) the angle of the vector a + b + c, and (c) the magnitude and (d) the angle of a - b + c? What are the (e) magnitude and (f) angle of a fourth vector such that (a + b) (c + d) =0?
28 28 3 Solved Problems 1. Three vectors a, b, and c each have a magnitude of 50 m and lie in an xy-plane. Their directions relative to the positive direction of the axis are 30, 195, and 315, respectively. What are (a) the magnitude and (b) the angle of the vector a + b + c, and (c) the magnitude and (d) the angle of a - b + c? What are the (e) magnitude and (f) angle of a fourth vector such that (a + b) (c + d) =0?
29 3 Solved Problems 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan Three vectors are given by a = 3.0i + 3.0j - 2.0k, b = -1.0i - 4.0j + 2.0k, and c = 2.0i + 2.0j + 1.0k. Find (a) a. (b x c), (b) a. (b + c), and (c) a x (b + c ).
30 30 3 Solved Problems 3. The three vectors in Figure have magnitudes a=3.00 m, b=4.00 m, and c=10.0 m and angle ϴ=30.0. What are (a) the x component and (b) the y component of a; (c) the x component and (d) the y component of b; and (e) the x component and (f) the y component of c? If c = pa + qb, what are the values of (g) p and (h) q?
31 31 3 Summary Scalars and Vectors Scalars have magnitude only Vectors have magnitude and direction Both have units! Vector Components Given by Related back by Adding Geometrically Obeys commutative and associative laws Eq. (3-5) Eq. (3-6) Eq. (3-2) Unit Vector Notation We can write vectors in terms of unit vectors Eq. (3-7) Adding by Components Add component-by-component Eqs. (3-10) - (3-12) Scalar Product Dot product Eq. (3-20) Eq. (3-22) Scalar Times a Vector Product is a new vector. Magnitude is multiplied by scalar. Direction is same or opposite. Cross Product Produces a new vector in perpendicular direction Direction determined by right-hand rule Eq. (3-3) Eq. (3-24)
32 3 Vectors 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 32 Additional Materials
33 3-2_4 Vectors and Their Components 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 33 Learning Objectives 3.01 Add vectors by drawing them in head-to-tail arrangements, applying the commutative and associative laws Subtract a vector from a second one Calculate the components of a vector on a given coordinate system, showing them in a drawing Given the components of a vector, draw the vector and determine its magnitude and orientation Convert angle measures between degrees and radians.
34 3-5_6 Unit Vectors, Adding Vectors by Components 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 34 Learning Objectives 3.06 Convert a vector between magnitude-angle and unitvector notations Add and subtract vectors in magnitude-angle notation and in unit-vector notation Identify that, for a given vector, rotating the coordinate system about the origin can change the vector's components, but not the vector itself.
35 3-8 Multiplying Vectors 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 35 Learning Objectives 3.09 Multiply vectors by scalars Identify that multiplying a vector by a scalar gives a vector, the dot product gives a scalar, and the cross product gives a perpendicular vector Find the dot product of two vectors Find the angle between two vectors by taking their dot product Given two vectors, use the dot product to find out how much of one vector lies along the other Find the cross product of two vectors Use the right-hand rule to find the direction of the resultant vector In nested products, start with the innermost product and work outward.
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