vectors day 2. scalar multiplication a vector can be multiplied by a real number multiplying a...
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VectorsDay 2
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Scalar Multiplication
• A vector can be multiplied by a real number• Multiplying a vector by a positive number changes its size, but not its
direction.• Multiplying a vector by a negative number changes its direction and its size
(unless it is multiplied by -1)
• The multiplication of a scalar, k, and a vector, v, is denoted as kv• A scalar “scales” the size of the vector.
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Adding vectors – “The Triangle Method”
• The process of geometrically adding two vectors is as follows:• Given vector v and vector u
1) Draw vector v2) At the terminal point of v, draw vector u3) Draw the resultant vector (r) from the initial point of v to the terminal point
of u
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Examples
• 1. v + u • 2. u + v
u
v
r
u
v
r
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Look!!!
u
v
r
u
v
r
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Example: Subtraction
• 4. u - v
u
v
r
v
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Adding vectors in component form
Given v = and w = 1. Geometrically add the vectors
• Find the component form of the resultant vector.
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Scalar Multiplication and Component Form
• Given a vector v = and a scalar k,kv= =
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Examples
• Given v = and u = , find each of the following: 1. v + u 2. 3u 3. 3u + 2v
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Unit vectors
• Unit vectors are vectors that have a magnitude of 1• The horizontal unit vector has
component form • The vertical unit vector has
component form
To find the unit vector of any non-vertical or non-horizontal vector:1. Find the magnitude of the vector2. Multiply the vector by the
reciprocal of its magnitude (basically divide the vector by its magnitude to give it a length of 1)
3. Perform the scalar multiplication on the appropriate form of the vector (the form the problem was written in)
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Examples
• Determine the unit vector that has the same direction as each of the following:
1. 2. 3.
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Example
• Write the vector with the given magnitude in the same direction as the vector given:
= 5
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Assignment #2
6.3 Exercises#13-22, 25-28,35-40, 45-46
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Examples