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10.2 Vectors and Vector Value Functions

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Quantities that we measure that have magnitude but not direction are called scalars. Quantities such as force, displacement or velocity that have direction as well as magnitude are represented by directed line segments. A B initial point terminal point The length is or AB

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A B initial point terminal point A vector is represented by a directed line segment. Vectors are equal if they have the same length and direction (same slope).

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A vector is in standard position (0) if the initial point is at the origin. x y The component form of this vector is:

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A vector is in standard position if the initial point is at the origin. x y The component form of this vector is: The magnitude (length) ofis:

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If P(p 1, p 2 ) and Q(q 1, q 2 ) are the initial and terminal points of a directed line segment, the component form of the vector v represented by is. This means the length (magnitude) of v is

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P Q (-3,4) (-5,2) The component form of is: v (-2,-2) A vector has an initial point at (-3, 4) and terminal point at (-5, 2).

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is the zero vector and has no direction. If v is a nonzero vector in plane, then the vector has length 1 and the same direction as v. u is called a unit vector in the direction of v.

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Ex. Find a unit vector in the direction of One of the vector operations says I can multiply a scalar by both components of a vector. We can make sure this is a unit vector by seeing if the magnitude = 1.

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Vector Operations: (Add the components.)

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v v u u u+v u + v is the resultant vector. (Parallelogram law of addition)

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Vector Operations: (Add the components.) (Subtract the components.)

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Vector Operations: Scalar Multiplication: Negative (opposite):

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The unit vectors and are called the standard unit vectors in the plane and are denoted by and These vectors can be used to represent any vector uniquely:

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Example: Let u be the vector with initial point (2, -5) and terminal point (-1, 3), and let v = 2i – j. Write w = 2u – 3v as a linear combination of i and j.

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