Geometry Lesson 8.7.notebook
April 18, 2016
VECTORS
Vector ­ a quantity that has both magnitude and direction.
Magnitude ­ the length of a vector.
Direction ­ the angle that is formed by the vector and the positive x ­axis or any other horizontal line.
Vectors are crucial for airplane pilots and flying planes and navigating boats too.
Vectors can be represented as directed line segments. The above vector can be called It has an initial point A and a terminal point B.
When placed on the coordinate plane, a vector is in standard position when it has its initial point at the origin.
Vector w is in standard position.
Component form ­ a vector is described in terms of horizontal change x and vertical change y from its initial point to its terminal point. The component form of
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Geometry Lesson 8.7.notebook
April 18, 2016
EXAMPLE: Write the component form of Try: Find the component form of Find the magnitude and direction of a vector.
Find the magnitude and direction of , if R(2,4)
and T(­3, ­2)
To find the magnitude of
use the distance formula. 2
Geometry Lesson 8.7.notebook
April 18, 2016
The x component (from ­3 to 2) is 5. The y component
(from ­2 to 4) is 6.
In order to find angle R, we have the opposite and adjacent, so use tangent.
If we were to move our initial point R to the origin, which quadrant would our terminal side be in? (Quadrant 3)
The direction of the vector is 180 + 50.2 = 230.2
Try: Find the magnitude and direction for , if F(­1,5) and G(3,­2) HINT: If y/x = 0 and x is positive, then tan­1 is 0. If x is negative then tan­1 is 180.
If y/x = undefined and y is positive then tan­1 is 90. If y is negative, then tan­1 is 270.
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Geometry Lesson 8.7.notebook
April 18, 2016
Equal vectors ­ two vectors are equal if and only if they have the same magnitude and direction.
Parallel vectors ­ two vectors are parallel if and only if they have the same or opposite directions.
(±180).
Opposite vectors ­ two vectors are opposite vectors if they have the same magnitude and
opposite directions.
Resultant ­ The sum of two vectors.
VECTOR ADDITION
Parallelogram Method
Step 1: Place both vectors at the same initial point.
Step 2: Complete the parallelogram. The resultant is the diagonal of the parallelogram.
Triangle Method
AKA tip to tail method
Step 1: Place the initial point of the second vector at the terminal point of the first.
Step 2: The resultant connects the initial point of the first vector and the terminal point of the second.
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Geometry Lesson 8.7.notebook
April 18, 2016
Examples on the white board: (use triangle method and parallelogram method)
Remember to make sure your vectors have the same magnitude and direction.
Find (a negative in front of a vector means you will take the opposite vector. Switch directions by 180)
Vectors can also be added or subtracted algebraically by adding or subtracting their horizontal and vertical components.
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Geometry Lesson 8.7.notebook
April 18, 2016
Write each vector in component form. Round to the nearest tenth.
a. First make a right triangle using the horizontal of the vector and use the angle supplementary to the angle of the vector. Then use SOHCAHTOA to find your x and y components.
b. 6
Geometry Lesson 8.7.notebook
April 18, 2016
Algebraic Vectors
Cheerleaders are using an air launcher to project t­shirts into the stands. The wind is blowing in the same direction that the shirts are launched. If a shirt leaves the launcher at 40 feet per second at an angle of and the wind is blowing 7 feet per second parallel to the horizontal, what is the resultant velocity and direction of the T­shirt?
Step 1: Draw a diagram to represent each direction of the vector.
Find the direction vector for the T­shirts in component form. Find x and y.
Use trig to find the vector's direction.
The component for the wind is The component vector for the T­shirts including the wind would be
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Geometry Lesson 8.7.notebook
April 18, 2016
Try: What would the resultant magnitude and direction of the shirt if the wind is blowing in the opposite direction?
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Geometry Lesson 8.7.notebook
April 18, 2016
P. 597 8­38
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