Vectors!
What is a vector?
Well, it's magnitude and direction represented by an arrow!
i, j and k are all unit vectors with length 1
i is in the x direction
j is in the y direction
k is in the z direction
Vector Notation:
In 3-Space, vector V starts at the origin and terminates at the point (Vx, Vy, Vz), therefore it has components <Vx, Vy, Vz>. Vx indicated the magnituse of the x-component of the vector, Vy the magnitude of the y-component and Vz the z-component.
<Vx, Vy, Vz> can be translated anywhere along the graph. In other words, two vectors are equal if and only if their corresponding components are equal.
Properties of Vectors:
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| When adding or subtracting vectors, simply add/subtract the components |
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| Figure 13.2.6: Subtracting vectors is the same as adding the negative vector |
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| Some other properties of Vectors. They essentially follow the same principles of everyday Math! |
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The resultant vector is always the sum of two vectors! |
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| Here's an example of manipulating vectors |
More Properties and Terms:
Finding a vector when given the initial point, P1, and the terminal point, P2:
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| In both 2 and 3 space, the components of vector P1P2 are found by subtracting P1 from P2 |
Using Laws of Sines and Cosines to Find the Magnitude and Angle of a Resultant Vector:
Throwback to Trig!! #TBTT :
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| Red indicates usage of the Law of Cosines. Blue indicates usage of the Law of Sines |
Here's an example!!
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Normalizing Vectors:
(Watch Out!
Finding the norm and normalizing
are NOT the same!!)
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| How to find the Norm of a vector (which is the same as the magnitude!) |
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| Normalizing: Creating a unit vector with the same angle as vector V |
Heres an example!
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| We normalized the vector, creating a unit vector with the components 2/3i + 2/3j - 1/3k
Here's Some More Helpful Links:
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