8.1 and 8.2 answers

8.3: Vectors February 9, 2009

Objectives Learn basic concepts about vectors. Perform operations on vectors. Learn and apply the dot product. Use vectors to calculate work.

The basics: Scalars vs. Vectors Scalars are quantities which are fully described by a magnitude (or numerical value) alone. Vectors are quantities which are fully described by both a magnitude and a direction.

More about vectors A vector quantity can be represented by a directed line segment. Vectors are equal if they have the same direction and magnitude. They are usually represented by a letter in boldface type, such as a, b, v, or F.

Describing vectors If the initial point is placed on the origin, then its terminal point (a 1, a 2 ) can bed used to determine v. a 1 is the horizontal component. a 2 is the vertical component.

Magnitude of a vector If v =, then its magnitude (or length) is: ||v|| =√ (a a 2 2 ) If ||v|| = 1, then v is a unit vector.

Vector Addition If a = and b =, the sum of a and b is… a+b = + =,

Vector Subtraction If a = and b =, the difference of a and b is… a - b = - =,

Scalar multiplication If v = and k is a real number, then the scalar product kv is kv= k =

Dot product If a = and b =, the dot product of a and b is… a b = a 1 b 1 + a 2 b 2

Angle between two vectors If a and b are two non-zero vectors, then the angle θ between a and b is given by θ = cos -1 (a b) (||a|| ||b||) Vectors are perpendicular if and only if a b =0.

Work If a constant force F is applied to an object that moves along a vector D, then the work W done is W=F D

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