Basic Math Vectors and Scalars Addition/Subtraction of Vectors Unit Vectors Dot Product.
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Basic Math Vectors and Scalars Addition/Subtraction of Vectors Unit Vectors Dot Product
Scalars and Vectors (1) Scalar – physical quantity that is specified in terms of a single real number, or magnitude Ex. Length, temperature, mass, speed Vector – physical quantity that is specified by both magnitude and direction Ex. Force, velocity, displacement, acceleration We represent vectors graphically or quantitatively: Graphically: through arrows with the orientation representing the direction and length representing the magnitude Quantitatively: A vector r in the Cartesian plane is an ordered pair of real numbers that has the form. We write r= where a and b are the components of vector v. Note: Both and r represent vectors, and will be used interchangeably.
Scalars and Vectors (2) The components a and b are both scalar quantities. The position vector, or directed line segment from the origin to point P(a,b), is r=. The magnitude of a vector (length) is found by using the Pythagorean theorem: Note: When finding the magnitude of a vector fixed in space, use the distance formula.
Operations with Vectors (1) Vector Addition/Subtraction The sum of two vectors, u= and v= is the vector u+v =. Ex. If u= and v=, then u+v= Similarly, u-v= =
Operations with Vectors (2) Multiplication of a Vector by Scalar If u= and c is a real number, the scalar multiple cu is the vector cu=. Ex. If u= and c=2, then cu= cu=
Unit Vectors (1) A unit vector is a vector of length 1. They are used to specify a direction. By convention, we usually use i, j and k to represent the unit vectors in the x, y and z directions, respectively (in 3 dimensions). i= points along the positive x-axis j= points along the positive y-axis k= points along the positive z-axis Unit vectors for various coordinate systems: Cartesian: i, j, and k Cartesian: we may choose a different set of unit vectors, e.g. we can rotate i, j, and k
Unit Vectors (2) To find a unit vector, u, in an arbitrary direction, for example, in the direction of vector a, where a=, divide the vector by its magnitude (this process is called normalization). Ex. If a=, then is a unit vector in the same direction as a.
Dot Product (1) The dot product of two vectors is the sum of the products of their corresponding components. If a= and b=, then a·b= a 1 b 1 +a 2 b 2. Ex. If a= and b=, then a·b=3+32=35 If θ is the angle between vectors a and b, then Note: these are just two ways of expressing the dot product Note that the dot product of two vectors produces a scalar. Therefore it is sometimes called a scalar product.
Dot Product (2) Convince yourself of the following: Conclusion: After you define the direction of an arbitrary vector in terms of the Cartesian system, you can find the projection of a different vector onto the arbitrary direction. By dividing the above equation by the magnitude of b, you can find the projection of a in the b direction (and vice versa).