Precalculus: 8-3 Dot Product and Vector Projections
Objectives:
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Find the dot product of two vectors, and use the dot product to find the angle between them.
Find the projection of one vector onto another.
Common Core State Standards: N-VM.4
Day 1:
Intro: Ever run out of gas? It’s work getting your car to a gas station if you have to
push it there. From a physics standpoint, work is the magnitude of a force
applied to an object multiplied by the distance through which the object
moves parallel to the applied force. Work, such as that done to push a car a
specific distance, can also be calculated using a vector operation called
a dot product.
Perpendicular Vectors
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In physics, torque is the measure of the effectiveness of a force in turning an object about a pivot point. We
can use perpendicular vectors to find the torque of a force.
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Perpendicular Vectors & Inner Products:
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1. Draw perpendicular vectors a ,b & AB .
2. What is the relationship between the magnitudes of each of these?
3. Rewrite the magnitudes of the components using coordinates.
4. Solve this equation for  AB  . Compare this to the relationship in step 2 above.
2
Dot Products
*The dot product of 2 vectors produces a scalar ( a real #).
****2 nonzero vectors are PERPENDICULAR if and only if their dot products = 0. (see front pg. of notes)
Dot Product Properties:
1. Commutative Prop.
2. Distributive Prop.
3. Scalar Multiplication Prop.
4. Zero Vector Dot Product Prop.
5. Dot Product and Vector Magnitude Relationship.
Could you determine this using another method?
Using the dot product to find the magnitude is the same as …..
Proof:
Day 1 HW: P. 506 #3-21 (odd)
Day 2: Vector Projection – Watch these videos 1st!
https://www.youtube.com/watch?v=3VlxQPeNJFs & https://www.youtube.com/watch?v=LOhSSWsT6vU
Ex 7. Mary sits on a sled on the side of a hill
inclined at 60  . What force is required to keep
the sled from sliding down the hill if the weight
of Mary and the sled is 125 pounds?
Ex 8. A person pushes a reel mower with a constant force of 40 newtons at a constant angle of 45 .
Find the work done in joules moving the mower 12 meters.
Ex 9. Rick is pushing a vacuum cleaner with a force of 85 pounds. The handle of the vacuum cleaner
makes a 60  angle with the floor. How much work in foot-pounds does he do if he pushes the
vacuum cleaner 6 feet?
Day 2 HW: P. 506 #27-31 (odd), Both 34’s, 35, 36