PHYS 101 Lecture 2 - Simon Fraser University
... © 2001 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. ...
... © 2001 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. ...
4.3.1) Yes, it is a subspace. It is clearly a subset of R2
... If (c1 − c3 ), (c3 − c2 ), and c2 are all zero, then c2 = 0, from which c3 = 0, and so c1 = 0. Since c1 , c2 , and c3 are not all zero, (c1 − c3 ), (c3 − c2 ), and c2 are not all zero, and so we have a nontrivial linear combination of w1 , w2 , and w3 which gives 0. Therefore, T is linearly dependen ...
... If (c1 − c3 ), (c3 − c2 ), and c2 are all zero, then c2 = 0, from which c3 = 0, and so c1 = 0. Since c1 , c2 , and c3 are not all zero, (c1 − c3 ), (c3 − c2 ), and c2 are not all zero, and so we have a nontrivial linear combination of w1 , w2 , and w3 which gives 0. Therefore, T is linearly dependen ...
Dot Product, Cross Product, Determinants
... A2 = (a2 b3 − a3 b2 )2 + (a3 b1 − a1 b3 )2 + (a1 b2 − a2 b1 )2 . Hence A = k~xk with the vector ~x defined by ~x := (a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ). Note that x j = ak b` − a` bk where 1. j, k, ` are different 2. k, ` are in “positive order” if we arrange 1,2,3 on a circle. This vec ...
... A2 = (a2 b3 − a3 b2 )2 + (a3 b1 − a1 b3 )2 + (a1 b2 − a2 b1 )2 . Hence A = k~xk with the vector ~x defined by ~x := (a2 b3 − a3 b2 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ). Note that x j = ak b` − a` bk where 1. j, k, ` are different 2. k, ` are in “positive order” if we arrange 1,2,3 on a circle. This vec ...
Handout #5 AN INTRODUCTION TO VECTORS Prof. Moseley
... segments equivalent to a given vector in G forms an equivalence class. Two directed line segments are related if they are in the same equivalence classes. This relation is called an equivalence relation since all directed line segments that are related can be tho ught of as being the sam e. Th e equ ...
... segments equivalent to a given vector in G forms an equivalence class. Two directed line segments are related if they are in the same equivalence classes. This relation is called an equivalence relation since all directed line segments that are related can be tho ught of as being the sam e. Th e equ ...
Vectors and Matrices – Lecture 2
... The scalar product gives a way to multiply two vectors and get a scalar. In three dimensions, there is another way to multiply two vectors which gives a vector. The vector product (or cross product) of a and b is written a × b. Provided a 6= 0 and b 6= 0 it has the following properties: it is a vect ...
... The scalar product gives a way to multiply two vectors and get a scalar. In three dimensions, there is another way to multiply two vectors which gives a vector. The vector product (or cross product) of a and b is written a × b. Provided a 6= 0 and b 6= 0 it has the following properties: it is a vect ...
The Dimension of a Vector Space
... a 1n c 1 a 2n c 2 a nn c n a mn c m 0. The above system is a homogeneous linear system with more unknowns (m) than equations (n). Such a system must have non–trivial solutions. Thus, there exist scalars c 1 , c 2 , c n , , c m , not all zero, that satisfy the above system. These ...
... a 1n c 1 a 2n c 2 a nn c n a mn c m 0. The above system is a homogeneous linear system with more unknowns (m) than equations (n). Such a system must have non–trivial solutions. Thus, there exist scalars c 1 , c 2 , c n , , c m , not all zero, that satisfy the above system. These ...
Special Factoring ( )( ) ( ) ( ) ( )( ) ( )( ) Converting Between Degree
... which quadrant this angle should be in and measure the angle counterclockwise from the positive x-axis. The horizontal component of the vector 〈 The vertical component of the vector 〈 ...
... which quadrant this angle should be in and measure the angle counterclockwise from the positive x-axis. The horizontal component of the vector 〈 The vertical component of the vector 〈 ...
Bivector
In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered an order zero quantity, and a vector is an order one quantity, then a bivector can be thought of as being of order two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions. They can be used to generate rotations in any dimension, and are a useful tool for classifying such rotations. They also are used in physics, tying together a number of otherwise unrelated quantities.Bivectors are generated by the exterior product on vectors: given two vectors a and b, their exterior product a ∧ b is a bivector, as is the sum of any bivectors. Not all bivectors can be generated as a single exterior product. More precisely, a bivector that can be expressed as an exterior product is called simple; in up to three dimensions all bivectors are simple, but in higher dimensions this is not the case. The exterior product is antisymmetric, so b ∧ a is the negation of the bivector a ∧ b, producing the opposite orientation, and a ∧ a is the zero bivector.Geometrically, a simple bivector can be interpreted as an oriented plane segment, much as vectors can be thought of as directed line segments. The bivector a ∧ b has a magnitude equal to the area of the parallelogram with edges a and b, has the attitude of the plane spanned by a and b, and has orientation being the sense of the rotation that would align a with b.