Vectors and Vector Multiplication. Vector quantities are those that have magnitude and direction, such as:. Displacement, x or Velocity, Acceleration, Force, Torque, Electric field, ….to name just a few. Scalar quantities have only magnitude:. Speed, v Distance, d Time, t
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Multiplication of scalar quantities follows all the “usual” rules, including:Distributive a(b+c) = ab + acCommutative ab = baAssociative (ab)c = a(bc)
Addition of vectors is commutative and associative and follows the subtraction rule:A+B = B+A(A+B)+C = A+(B+C)A-B = A+(-B)
However rules:, multiplication of vectors has a new set of rules—the vector cross product (or “vector product”) and the vector dot product or “scalar product”.
Vector Dot Product rules:or Scalar ProductA·B = AB cosEssentially, this means multiplying the first vector times the component of the second vector that is in the same direction as the first vector—yielding a product that is a scalar quantity.
A·B = AB cos
Multiple the magnitude of vector A times the magnitude of vector B times the cosine of the angle between them—or multiply the components that are in the same direction. The answer is a scalar with the units appropriate to the product AB.
Vector Cross Product rules:or Vector ProductAxB = AB sinEssentially, this means multiplying the first vector times the component of the second vector that is perpendicular to the first vector—yielding a product that is a vector quantity. The direction of the new vector is found using the right hand rule.
Multiple the magnitude of vector A times the magnitude of vector B times the sine of the angle between them—or multiply the components that are perpendicular. The answer is a vector with the units appropriate to the product AB and direction found by using the right hand rule.
F = q (vxB)
where q is the charge on a proton, v is 3x105 m/s to the left on the paper, and B is 500 N/C outward from the paper toward you. The equation for this is also: F = qvB sin
The answer for the force is 2.4 x 10 rules:-11 newtons toward the top of the paper.
Unit vectors have a size of “1” but also have a direction that gives meaning to a vector.
We use the “hat” symbol above a unit vector to indicate that it is a unit vector.
For example, is a vector that is 1 unit in the x-direction. The quantity 6 meters is a vector 6 meters long in the x-direction.
What is the work done by a force of 6 newtons east on an object that is displaced 2 meters east?
What is the work done by a force of 6 newtons east on an object that is displaced 2 meters north?
What is the work done by a force of 6 newtons east on an object that is displaced 2 meters at 30 degrees north of east?