Vectors: Definitions, Properties, and Applications
Explore the world of vectors with examples, definitions, and properties such as addition, scalar multiplication, and projection. Understand how vectors represent translations and grasp concepts like commutativity and distributive properties in vector operations.
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Presentation Transcript
Examples of vectors and relations between them Equal vectors Different directions Ending point (head) S Q a -a a b c Starting point (tail) R o P Null vector Free vector Equivalent bound vectors Opposite orientation (same attitude) Different length (magnitude) represent
Interpretation of a vector as a translation of a rigid body ? ? Q ? ? R ? Q ? ?? = ?? = ?? = ?
Commutativity of vector addition ? ? ? ? a a + b b = b b + a a
Associativity of vector addition ? ? ? (a a + b b) + c + c = a a + (b + c b + c)
Multiplication of a vector by a scalar ? 2? 2? 1 2?
Distributive property ?? ? ?? ? ? ? ? ? ? p(a a + b b) = pa a + pb b also (p+q)a a = pa a + qa a
Definition of vector difference ? ? ? ? ? ? -? a a - b b = a a + (-1)b b
Projection of a vector on a line ? ? ? ?? ? ? P
Distributive property ? ? ? ?? ? ? (? + ?)? ? ? + ? ? ?? ? P (a a + b b)l = a al + b bl
Calculating projection length ? ? ?? ? ??= ?? = |?|cos?
Cross product (a) ? ? ? = ? ? sin? ? ? ? ? (b) ? ? ? ? = ? ? ? ? ? ? ? ? = ? ?
Toward a distributive property ? ? ? ? ? + ? (? + ?) ? ?
The distributive property ? ? + ? ? = (? + ?) ? ? ? + ? ?= (? + ?) ? ? a ? + ? ? = ? + ? ? ? ? ? ? ? ?, ?
Vector in coordinates a a = ax ? + ay ? + az ?
