11.3 the dot product. geometric interpretation of dot product a dot product is the magnitude of one...

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11.3 The Dot Product

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Page 1: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

11.3 The Dot Product

Page 2: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Page 3: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Geometric interpretation of dot product

A dot product is the magnitude of one vector times the portion of the vector that points in the same direction as that vector (the projection in the direction of the other.)

Page 4: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Find the dot product of the given vectors

• u = (4,10) v = (-2,3)

• u = (1,5,7) v = (-1,3,4)

Note: To find a dot product on the TI89

Press 2nd 5 (math) – 4 matrix – L Vector ops – 3 dotP

dotP([1,5,7],[-1,3,4])

Page 5: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Note property 5 is explained on the next slide.

Page 6: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

__We can see that using the Pythagorean Theorem yields the same result as √a∙a

So we can write the length or magnitude of a vector in terms of the dot product. This will be important in the second semester.

Page 7: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Angle Between Two Vectors

Proven on last slide

Page 8: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Alternative form of dot product

Page 9: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Notes: This definition will allow us to expand the notion of orthogonal to higher Dimensions. (This will be important next semester in Linear Algebra.)

Orthogonal and perpendicular are generally used interchangeably. However there is a subtle difference. Perpendicular means that two items (planes, lines segments vectors … whatever) must meet to make a 90 degree angle… However, orthogonal includes this situation plus includes the zero vector is orthogonal to all other vectors even though we could not say that it is perpendicular to all other vectors.

Page 10: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Find the angle between u and v

• u = (3,-1,3)

• v =(-4,0,2)

Page 11: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

What is meant by the angle between two vectors?

Page 12: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Determine if the given vectors are orthogonal

• u = (3,-1,2)

• w =(1,-1,3)

Page 13: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Determine if the given vectors are orthogonal

• u = (3,-1,2)

• w =(1,-1,3)

u and v are not orthogonal because the dot product is not 0.

What value x will make vectors u and q orthogonal?

q = (1,-1,x)

Page 14: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Page 15: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Example 5

Find the projection of u onto v and the vector component of u orthogonal to v

• u = 3i – 5j +2k v = 7i + j -2k

Page 16: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Page 17: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Note solve this problem 3 ways:Solve with special right trianglesSolve with Trigonometry and force times distanceSolve with the dot product

Page 18: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

Page 19: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

"A mathematician is a device for turning coffee into theorems“ -- P. Erdos

Page 20: 11.3 The Dot Product. Geometric interpretation of dot product A dot product is the magnitude of one vector times the portion of the vector that points

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