8.1 and 8.2 answers. 8.3: vectors february 9, 2009
TRANSCRIPT
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8.1 and 8.2 answers
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8.3: Vectors
February 9, 2009
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Objectives
• Learn basic concepts about vectors.• Perform operations on vectors.• Learn and apply the dot product.• Use vectors to calculate work.
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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.
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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.
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Describing vectors
If the initial point is placed on the origin, then its terminal point (a1, a2) can bed used to determine v.
a1 is the horizontal component.
a2 is the vertical component.
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Magnitude of a vector
If v = <a1, a2>, then its magnitude (or length) is:
||v|| =√ (a12 + a2
2)
If ||v|| = 1, then v is a unit vector.
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Vector Addition
If a = < a1, a2> and b = < b1, b2>, the sum of a and b is…
a+b = < a1, a2> + < b1, b2>
= < a1+ b1, a1+ b2>,
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Vector Subtraction
If a = < a1, a2> and b = < b1, b2>, the difference of a and b is…
a - b = < a1, a2> - < b1, b2>
= < a1- b1, a1- b2>,
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Scalar multiplication
If v = < v1, v2> and k is a real number, then the scalar product kv is
kv= k < v1, v2>
= < kv1, kv2>
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Dot product
If a = < a1, a2> and b = < b1, b2>, the dot product of a and b is…
a • b = a1b1 + a2b2
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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.
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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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assignment
• Pages 687-8• 25-34• 51-62