VectorsScalar QuantitiesVector QuantitiesVector DiagramVector NotationVectors in EquationsGraphical AdditionParallelogramCommutative PropertyParallel and AntiparallelCancellationVectorsVectorsScalar QuantitiesScalar QuantitiesAny measurement that consists of a single number is Any measurement that consists of a single number is a a scalarscalar..•72 72 °°FF•500 milliliters500 milliliters•1.5 volts1.5 volts•6 hours, 24 minutes (consider this as 6.4 h)6 hours, 24 minutes (consider this as 6.4 h)Most measured quantities consist of only a single Most measured quantities consist of only a single magnitude.magnitude.Vector QuantitiesVector QuantitiesA measurement that requires more than one value to A measurement that requires more than one value to describe it is a describe it is a vectorvector..•10 km to the northeast10 km to the northeast•At 41.9At 41.9°° N latitude and 88.7 N latitude and 88.7° W longitude° W longitude•15 pounds of force directed down15 pounds of force directed downThese quantities can be thought of as carrying the These quantities can be thought of as carrying the value from a starting point to a destination.value from a starting point to a destination.The word The word vectorvector means carrier. means carrier.Vector DiagramVector DiagramOne representation of a One representation of a vector is an arrow.vector is an arrow.The tail shows the start of The tail shows the start of the vector.the vector.The tip points in the The tip points in the direction.direction.The length of the arrow The length of the arrow shows the magnitude.shows the magnitude.tail: starttip: directionlength: magnitudeVector NotationVector NotationA vector variable is A vector variable is represented by a small arrow represented by a small arrow over the top of the variable.over the top of the variable.Some texts use Some texts use boldfaceboldface for for vectors, but that can be hard vectors, but that can be hard to distinguish on some to distinguish on some backgrounds. Our text uses backgrounds. Our text uses both boldface and an arrow.both boldface and an arrow.The magnitude of a vector is The magnitude of a vector is a scalar.a scalar.It is represented as the It is represented as the absolute value of the vector, absolute value of the vector, or just the variable without or just the variable without the vector symbol.the vector symbol.AvAA Vectors in EquationsVectors in EquationsVector variables can be used in equationsVector variables can be used in equations•For instance, For instance, Vector variables are different from scalar variablesVector variables are different from scalar variables•They are different dimension, They are different dimension, CBAxX LxX Graphical AdditionGraphical AdditionThe two vectors can be added The two vectors can be added graphically.graphically.The tail of the second vector is The tail of the second vector is placed at the tip of the first.placed at the tip of the first.The length and directions are The length and directions are kept the same.kept the same.The result is the total vector. Its The result is the total vector. Its magnitude can be measured magnitude can be measured on the graph.on the graph.A = 2.0B = 3.0C = 4.6CBAParallelogramParallelogramForce vectors act on a common object at a single Force vectors act on a common object at a single point.point.If two vectors are added from a common origin one If two vectors are added from a common origin one can be shifted to make a parallelogram.can be shifted to make a parallelogram.This is the same as putting the tail to the tip.This is the same as putting the tail to the tip.CBAA = 2.0 NB = 3.0 NC = 4.6 NCommutative PropertyCommutative PropertyVectors can be shifted as long as they don’t change Vectors can be shifted as long as they don’t change direction and magnitude.direction and magnitude.Vectors can be added in reverse order and get the Vectors can be added in reverse order and get the same result.same result.CABCBAParallel and AntiparallelParallel and AntiparallelVectors that point in the Vectors that point in the same direction are same direction are parallelparallel..Vectors that point in opposite Vectors that point in opposite directions are directions are antiparallelantiparallel..CancellationCancellationWhat happens if we add two antiparallel vectors of What happens if we add two antiparallel vectors of equal magnitude?equal magnitude?The vector sum is a zero length vector. The vectors The vector sum is a zero length vector. The vectors cancel out.cancel out.nextAA0)(
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