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- https://phys.libretexts.org/Courses/Tuskegee_University/Algebra_Based_Physics_I/03%3A_Two-Dimensional_Kinematics/3.04%3A__Vector_Addition_and_Subtraction-_Analytical_MethodsAnalytical methods of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, be...Analytical methods of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. However, analytical methods are more concise, accurate, and precise than graphical methods, which are limited by the accuracy with which a drawing can be made.
- https://phys.libretexts.org/Courses/Coalinga_College/Physical_Science_for_Educators_(CID%3A_PHYS_14)/09%3A_Motion/9.02%3A_Introduction-_Fundamentals_of_Motion-_Scientific_Overview/9.2.04%3A_Vector_AdditionThe sum of the two vectors is the vector that begins at the origin of the first vector and goes to the ending of the second vector, as shown below. The resultant, or sum, vector would be the vector fr...The sum of the two vectors is the vector that begins at the origin of the first vector and goes to the ending of the second vector, as shown below. The resultant, or sum, vector would be the vector from the origin of the first vector to the arrowhead of the last vector; the magnitude and direction of this sum vector would then be measured.
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Dourmashkin)/03%3A_Vectors/3.03%3A_VectorsFrom the physicist’s point of view, we are interested in representing physical quantities such as displacement, velocity, acceleration, force, impulse, and momentum as vectors. We must always understa...From the physicist’s point of view, we are interested in representing physical quantities such as displacement, velocity, acceleration, force, impulse, and momentum as vectors. We must always understand the physical context for the vector quantity. Thus, instead of approaching vectors as formal mathematical objects we shall instead consider the following essential properties that enable us to represent physical quantities as vectors.
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Introductory_Physics_II_(1112)/03%3A_Electric_Charge_and_Electric_Field/3.06%3A_Electric_Field_LinesNote that the electric field is defined for a positive test charge \(q\), so that the field lines point away from a positive charge and toward a negative charge. (Figure \(\PageIndex{2}\)) The electri...Note that the electric field is defined for a positive test charge \(q\), so that the field lines point away from a positive charge and toward a negative charge. (Figure \(\PageIndex{2}\)) The electric field strength is exactly proportional to the number of field lines per unit area, since the magnitude of the electric field for a point charge is \(E=k|Q|/r^{2}\) and area is proportional to \(r^{2}\).
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/66%3A_Appendices/66.16%3A_Vector_Arithmeticwhere \(\mathbf{i}\) is a unit vector (a vector of magnitude 1) in the \(x\) direction, \(\mathbf{j}\) is a unit vector in the \(y\) direction, and \(\mathbf{k}\) is a unit vector in the \(z\) directi...where \(\mathbf{i}\) is a unit vector (a vector of magnitude 1) in the \(x\) direction, \(\mathbf{j}\) is a unit vector in the \(y\) direction, and \(\mathbf{k}\) is a unit vector in the \(z\) direction. \(A_{x}, A_{y}\), and \(A_{z}\) are called the \(x, y\), and \(z\) components (respectively) of vector \(\mathbf{A}\), and are the projections of the vector onto those axes.
- https://phys.libretexts.org/Bookshelves/University_Physics/Radically_Modern_Introductory_Physics_Text_I_(Raymond)/02%3A_Waves_in_Two_and_Three_Dimensions/2.01%3A_Math_Tutorial__VectorsThe tail of vector B is collocated with the head of vector A, and the vector which stretches from the tail of A to the head of B is the sum of A and B, called C in Figure \(\PageIndex{1}\):. The quant...The tail of vector B is collocated with the head of vector A, and the vector which stretches from the tail of A to the head of B is the sum of A and B, called C in Figure \(\PageIndex{1}\):. The quantities \(A_{x}, A_{y}\) etc., represent the Cartesian components of the vectors in Figure \(\PageIndex{1}\):. A vector can be represented either by its Cartesian components, which are just the projections of the vector onto the Cartesian coordinate axes, or by its direction and magnitude.
- https://phys.libretexts.org/Bookshelves/College_Physics/College_Physics_1e_(OpenStax)/03%3A_Two-Dimensional_Kinematics/3.03%3A__Vector_Addition_and_Subtraction-_Analytical_MethodsAnalytical methods of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, be...Analytical methods of vector addition and subtraction employ geometry and simple trigonometry rather than the ruler and protractor of graphical methods. Part of the graphical technique is retained, because vectors are still represented by arrows for easy visualization. However, analytical methods are more concise, accurate, and precise than graphical methods, which are limited by the accuracy with which a drawing can be made.
- https://phys.libretexts.org/Bookshelves/College_Physics/College_Physics_1e_(OpenStax)/03%3A_Two-Dimensional_Kinematics/3.05%3A_Addition_of_VelocitiesVelocities in two dimensions are added using the same analytical vector techniques. Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies dramati...Velocities in two dimensions are added using the same analytical vector techniques. Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies dramatically with reference frame. Relativity is the study of how different observers measure the same phenomenon, particularly when the observers move relative to one another. Classical relativity is limited to situations where speed is less than about 1% of the speed of light (3000 km/s).
- https://phys.libretexts.org/Bookshelves/College_Physics/College_Physics_1e_(OpenStax)/18%3A_Electric_Charge_and_Electric_Field/18.05%3A_Electric_Field_Lines-_Multiple_ChargesDrawings using lines to represent electric fields around charged objects are very useful in visualizing field strength and direction. Since the electric field has both magnitude and direction, it is a...Drawings using lines to represent electric fields around charged objects are very useful in visualizing field strength and direction. Since the electric field has both magnitude and direction, it is a vector. Like all vectors, the electric field can be represented by an arrow that has length proportional to its magnitude and that points in the correct direction. (We have used arrows extensively to represent force vectors, for example.)
- https://phys.libretexts.org/Courses/Tuskegee_University/Algebra_Based_Physics_I/03%3A_Two-Dimensional_Kinematics/3.06%3A_Addition_of_VelocitiesVelocities in two dimensions are added using the same analytical vector techniques. Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies dramati...Velocities in two dimensions are added using the same analytical vector techniques. Relative velocity is the velocity of an object as observed from a particular reference frame, and it varies dramatically with reference frame. Relativity is the study of how different observers measure the same phenomenon, particularly when the observers move relative to one another. Classical relativity is limited to situations where speed is less than about 1% of the speed of light (3000 km/s).
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/09%3A_Vectors/9.03%3A_Vector_Arithmetic-_Algebraic_MethodsThen, recalling the rules for the multiplication of a vector by a scalar, A x i is a vector pointing in the x-direction, and whose length is equal to the projection A x . Similarly, A y j is a vector ...Then, recalling the rules for the multiplication of a vector by a scalar, A x i is a vector pointing in the x-direction, and whose length is equal to the projection A x . Similarly, A y j is a vector pointing in the y-direction, and whose length is equal to the projection A y . Then by the parallelogram rule for adding two vectors, vector A is the sum of vectors A x i and A y j (Fig. \(\PageIndex{1}\)).
