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The Geometric Representation of Vectors - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

When introduced to vectors for the first time, learning the geometric representation of vectors can help students understand their significance and what they really mean. The geometric representation of vectors can be used for adding vectors and can frequently be used to represent forces and find their resultant. The algebraic representation is used for more complex calculations.

What is a vector? Quantities like length, mass and temeperature that have magnitude but not direction are called scalars but quantities like velocity, acceleration and force that have magnitude and direction are called vectors. Now in Mathematics a vector is represented Geometrically by a directed line segment like this one ab and this is the notation for a vector notice the sort of half arrow over the top of them it's not the same as the notation for a ray which is a full arrow, so that would be a ray this would be vector. Notice also the order of the points is very important a is the initial point b is the terminal point when you list a vector you always start with the initial point and you finish with the terminal point the point where the arrow head goes. We also have a notation for the magnitude or length of a vector it looks like absolute value but you can read as absolute value or magnitude of vector ab.
Now two vectors are considered equal if they have the same length and the same direction so looking at these vectors, which ones are equal? Well these two have the same length but they have opposite direction so they're not considered equal. This one will be called vector ab, vector ab equals vector v because they have the same length and direction. Another vector that has the same length is vector w but again it's a different direction so it's a different vector. Now again it's very important that you know when you're naming a vector you have to name it with either two letters one denoting the initial point the other denoting the the terminal point or you can call it by one letter name usually a lower case letter like v or w but when you're handwriting the name of the vector you always put this half arrow over the top in your textbooks you may see a bold face letter v for vector v but since it's hard to write bold face by hand usually go with the arrow, so this vector by the way will be called vector fe right because the initial point is f and the terminal point is e so just remember a vector is represented by a directed line segment you can name a vector by using its endpoints first its initial point then its terminal point or you can give the whole vector a single lower case letter name but always with the half arrow over the top and remember two vectors are equal if they have the same length and same direction they can be in different locations but they have the same length and same direction they're the same vector.

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