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Vectors

Explore the differences between vector and scalar quantities in various measurements, including examples and notations used. Learn how vectors are represented and added graphically, understanding concepts like parallel and antiparallel vectors.

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Vectors

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  1. Vectors

  2. Scalar Quantities • Any measurement that consists of a single number is a scalar. • 72 °F • 500 milliliters • 2.54 centimeters • 6 hours, 24 minutes (consider this as 6.4 h) • Most measured quantities consist of only a single magnitude.

  3. Vector Quantities • A measurement that requires more than one value to describe it is a vector. • 10 km to the northeast • At 41.9° N latitude and 88.7° W longitude • 15 pounds of force directed down • These quantities can be thought of as carrying the value from a starting point to a destination. • The word vector means carrier.

  4. One representation of a vector is an arrow. The tail shows the start of the vector. The tip points in the direction. The length of the arrow shows the magnitude. Vector Diagram tip: direction length: magnitude tail: start

  5. A vector variable is represented by a small arrow over the top of the variable. Some texts use boldface for vectors, but that can be hard to distinguish on some backgrounds. Our text uses both boldface and an arrow. The magnitude of a vector is a scalar. It is represented as the absolute value of the vector, or just the variable without the vector symbol. Vector Notation

  6. Vectors in Equations • Vector variables can be used in equations • For instance, • Vector variables are different from scalar variables • They are different dimension,

  7. The two vectors can be added graphically. The tail of the second vector is placed at the tip of the first. The length and directions are kept the same. The result is the total vector. Its magnitude can be measured on the graph. Graphical Addition C = 4.6 B = 3.0 A = 2.0

  8. Parallelogram • Force vectors act on a common object at a single point. • If two vectors are added from a common origin one can be shifted to make a parallelogram. • This is the same as putting the tail to the tip. C = 4.6 N B = 3.0 N A = 2.0 N

  9. Commutative Property • Vectors can be shifted as long as they don’t change direction and magnitude. • Vectors can be added in reverse order and get the same result.

  10. Vectors that point in the same direction are parallel. Vectors that point in opposite directions are antiparallel. Parallel and Antiparallel

  11. Cancellation • What happens if we add two antiparallel vectors of equal magnitude? • The vector sum is a zero length vector. The vectors cancel out. next

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