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6.4 Vectors and Dot Products.notebook
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April 03, 2014
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6.4 Vectors and Dot Product
Plan:• Find the dot product of two vectors and use properties of the dot product.• find angles between vectors and determine whether two vectors are orthogonal.• Write vectors as sums of two vector components.• Use vectors to find the work done by a force.
Homework:
P 445 #3-27 (by 3‛s), 30, 31, 33-66 (by 3‛s)
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Dot Product (Vector Multiplication)
Given: u = and v =
then: this is a scalar (numeric) not a vector!
Examples: find the dot product.
#4 u = 3i + 2j v = -2i + j
#10 use find:
6.4 Vectors and Dot Products.notebook
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Properties of Dot Product
1.
2.
3.
4.
5.
Example #14 Use the dot product to find the magnitude of: u = 6i - 10j
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Angle Between Two VectorsIf θ is the angle between two nonzero vectors u and v, then
Examples:
#20 u = 2i - 3j v = i - 2j
Find the angle between the vectors.
#23
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Alternate form of Dot ProductIf then we find see that
is another way to write a dot product.
Orientations of two vectors:
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Orthogonal VectorsPerpendicular Vectors
u is orthogonal to v if
#35
Determine whether u and v are orthogonal, parallel or neither
#38 u = 8i + 4j v = -2i - j
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Group Challenge#43 Find the value of k so that the vectors u and v are orthogonal
u = -3ki + 2jv = -6i
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Find Vector Component using ProjectionsMany applications in physics and engineering requires that we decompose a vector into the sum of two vector components.
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Vector Components
u=w 1
+w2
w1w2
v
w1 and w2 are vector components of u
u=w1 +w
2
w1
w 2
v
w1 is the projection of u onto v
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Projection of u onto v
Decomposing a vector into components using projections.#47 Find projection of u on v, then write u as the sum of two orthogonal vectors one of which is proj vu.
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#46 Find projection of u on v, then write u as the sum of two orthogonal vectors of which one is proj vu.
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#53 Find two vectors in opposite directions that are orthogonal to the vector u. (There are many correct answers.)
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Finding Force - An application#62 Braking Load A SUV with a gross weight of 5400 pounds is parked on a slope of 10 o. Assume that the only force to overcome is the force of gravity. Find the force required to keep the vehicle from rolling down the hill. Find the force perpendicular to the hill.
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WorkWork is a force (as a vector) applied to an object in a specific direction. It is a scalar (numeric value)
Work Force asa vector
Distance as adirected line segment (vector)
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Examples#58 Find the work done in moving a particle from P to Q if the magnitude and direction of the force are given by v.
v = -2i + 3j • Write the directed line segment as a vector• Find the dot product• Label answer as distance/mass
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Examples#64 Work A force of 45 pounds in the direction of 30 o above the horizontal is required to slide a table across a floor. Find the work done if the table is dragged 20 feet .
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Examples#67 Work A toy wagon is pulled by exerting a force of 20 pounds on a handle that make a 25 o angle with the horizontal. Find the work done in pulling the wagon 40 feet .
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What is known about Θ, the angle between two nonzero vectors u and v, if each of the statement to the left are true:
Discuss your response with your shoulder partner and then be prepared to share you conclusions.
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