10.1.1 introduction to vectors today you will use geometry to define and perform operations using...
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10.1.1 Introduction to Vectors
Today you will use geometry to define and perform operations using
vectors. You will also write vectors in component form.
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An Introduction to Vectors• When we describe physical phenomenon such
as wind or a current in a river, we look at the direction of the force and its strength.
• These properties can be described by using vectors.
• Vectors are used to describe a wide variety of real world forms such as wind, velocity, and force because they have both magnitude and direction.
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Simon Says Vector Game
• Defining our space: North South East West
• One partner will “act out” the problem while the other partner will record the motions on the graph paper.
• Halfway through you will change roles with your partner.
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Simon Says Vector Game #1
• Go three steps east.• Go three steps south.• Go one step west.
• Go two steps south.• Go four steps west.• Go two steps north.
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Simon Says Vector Game #2
• Go two steps north.• Go three steps west.• Go one step east.
• Go three steps south.• Go four steps east.• Go one step south.• Go four steps west.• Go one step south.
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Shall We Dance?
• In the Simon Says activity, if the steps were followed correctly, did each person make the same movements?
• Was everyone in the same location?
• With your partner, complete the Sketch column of the Vector Line Dance Activity. Start each step at the ending point of the previous step.
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Vector Line Dance
• The vector represents the movement in the horizontal and vertical directions (x and y).
• The angle is the measurement from the
positive x-axis (this is called the standard angle).
• The length of the vector is called the magnitude.
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Vectors
The arrows you have been drawing are called vectors.
4 steps
4 steps
• Do these two vectors represent the same instruction?
• Are the starting points the same?
• Two vectors that represent the same instruction are called equivalent.
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Magnitude & Direction
• Vectors have both a length called magnitude and a direction. Looking at the instructions in the activity; find vectors that are equivalent in magnitude, but go in opposite directions.
Step 1 and Step 5
Step 4 and Step 7
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Angle and Magnitude
• Find the Angle and Magnitude for each step on the resource page.
• To find the angle, think of the starting point for each vector as the origin and figure out the angle to the vector from the positive horizontal axis.
• The Magnitude is the length of the vector. Draw a right triangle and use the Pythagorean theorem if necessary.
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Vector Notations
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Component Form
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Component Form
•With your partner, find the component form of
each vector on the resource page.
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Equivalent Vectors
• In the graphs below, vectors p and q are called equivalent vectors. Equivalent vectors are vectors that have the same magnitude and direction.
• Name two other pairs of equivalent vectors
• Draw a vector equivalent to x. Are everyone’s answers equivalent?
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Adding Vectors Geometrically• Copy vectors r and m on a sheet
of graph paper.• Step 1: Draw a vector
equivalent to m whose initial point coincides with the endpoint of r.
• Step 2: Draw a vector from the initial point of r to the endpoint of m.
• Step 3: Label this vector r + m.
We call this new vector the sum, or resultant, of r and m. This is called the head to tail method for adding vectors.
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Adding Vectors Geometrically• Draw another vector
labeled a which is equivalent to r + m.
• Copy vector p onto your graph paper. Then draw r + p. Label the resultant vector b.
• Find the component forms for vectors r, p, and b.
Using your observations, make a conjecture about adding vectors in component form. Test your conjecture by using your results from adding r and m.
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More Adding Vectors
In the diagram above, vector v is added to vector u (not shown) to get the resultant vector w. What is the component form of u?
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Assignment
HW 10.1.1
10-6 to 10-14
(pg. 476)