Notes on vectors

Vectors

Vectors:

Properties of the Three-Dimensional Universe

• There are 3 directions in space which are independent of one another.
• To specify the location of an object in space uniquely, it takes a minimum of three numbers.
• No matter which direction one chooses, there are only 3 directions in space that are all at 90^o to each other.

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SCALARS:

• Quantities that need only one number to uniquely specify their value. They are "one-dimensional" quantities.
• Quantities that can be "added" using simple math operations. Simple math operations (+, -, x, /) are "one-dimensional" operations only.

Examples: Mass, Time, Energy, Work, Power

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VECTORS:

• Quantities that are three-dimensional.
• Quantities that have direction as well as magnitude.
• Quantities that need three values to uniquely characterize them, one for each independent direction in space. ( In this course we are often able to reduce problems to a two-dimensional problems. We can do this since there are many situations in which the motion takes place in a plane.)

Examples: Location, Velocity, Acceleration, Force, Pressure

Vector Components:
There are two ways to express a two-dimensional vector.

• One is to state the magnitude and direction (A, q) of the vector.

• The second is to state the horizontal and vertical components (A_x, A_y) of the vector.

The components of a vector and its magnitude form a right triangle with the hypotenuse equal to the magnitude of the vector.

Addition of Two Vector:

• To add two vectors, the best method is to add their components.

Two-Dimensional Vector Sum:

Resolve each vectors in to their Components and Add the Components	Use the right triangle rules to determine the magnitude and direction of the vector sum.

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• For Vectors that are not in the same direction, you cannot simply add (or subtract) their lengths to find the length of their vector sum.
• The simple math operations (+ , - ) are "one-dimensional" actions and you can only add (or subtract) quantities that are in the same direction.
• The components of a vector that are along the same direction can be added (or subtracted).

Find the magnitude of the vector R that is the sum of the three vectors A , B , and C in the figure

Find the direction of the vector R.

Find the magnitude of the vector S = C – A – B

Find the direction of the vector S = C – A – B.

Law of Cosines:

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Law of Sines:

Length/Magnitude of a Vector:
The Dot Product of a vector with itself is always equal to its magnitude squared since q = 0^o and cos 0^o = 1.

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Unit Vectors:

• Vectors that indicate Direction Only.
• The Length of a Unit Vector is always equal to One.

The tag is used to denote a vector of length one in the same direction as the vector .

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_{Cartesian Expression for an Arbitrary Vector:}

Vector Components of	Numerical Components of

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