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Todays Plan C1 Marks – Class Average: 76% New Lesson –Midpoint (6.4) –Division Point (6.4) –Scalar Product (6.5) Homework Questions PRACTICE

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Midpoint of a Line Segment The coordinate of M are: M B A

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Properties of Midpoint of a Line Segment 1.MA + MB = 0 2.AM = MB 3.AM = ½AB 4.AB = 2AM M B A

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Division point of a Line Segment Point P divides the line: –in a 3:2 ratio or 3/2 from A or –in a 2:3 ration or 2/3 from B P B A 2 3

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Division point of a Line Segment We can also say: P is located at 3/5 from point A or P is located at 2/5 from point B P B A 2 3

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Division point of a Line Segment The vector equality is: AP = 3/5 AB or BP = 2/5 BA P B A 2 3

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Example 1 Consider A(1,4) and B(7,1), find P if AP = AB AB = (7-1,1-4) = (6,-3) AP = (x P -x A, y P -y A ) We know that AP = AB so fill in the other info… (x P - 1, y P - 4) = (6,-3)

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Example 1 Consider A(1,4) and B(7,1), find P if AP = AB (x P - 1, y P - 4) = (6,-3) (x P - 1, y P - 4) = (4,-2) Look at x and y individually… x P – 1 = 4 and y P – 4 = -2 x P = 4 +1 and y P = -2 +4 x P = 5 and y P = 2 So point P is (5,2)

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SCALAR PRODUCT The scalar product of two vectors u and v is a real number defined by: uv = ||u||×||v||×cosӨ u Ө v

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SCALAR PRODUCT 1.When vectors u and v are orthogonal, their scalar product is ZERO 2.uu = ||u||×||u||×cos0˚ = ||u|| 2

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Example 2 Consider u and v form a 30˚ angle and that ||u||=4 and ||v||=3. Find the scalar product. uv = 4 x 3 x cos30 uv = 10.39

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SCALAR PRODUCT IN THE CARTESIAN PLANE Let u=(a,b) and v=(c,d), we have: uv = ac + bd If we know the angle Ө formed by vectors u and v, we have: cosӨ = uv _ ||u|| ||v||

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Example 3 Let u=(4,1) and v=(2,3). Find the scalar product and the angle between the vectors uv = 4x2 + 1x3 uv = 11 What are ||u|| and ||v||? ||u||=4 2 +1 2 = 17 ||v|| = 2 2 +3 2 = 13 cosӨ = uv _ ||u|| ||v||

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Example 3 Let u=(4,1) and v=(2,3). Find the scalar product and the angle between the vectors cosӨ = uv _ ||u|| ||v|| cosӨ = 11 _ = 11 = 0.74 17 x 13 221 So Ө = cos -1 (0.74) Ө = 42.3˚

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PROPERTIES OF THE SCALAR PRODUCT uv = vu rusv = (rs)uv u(v+w) = uv + uw

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Last homework questions? (not the quiz questions…)

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HOMEWORK p. 300 #25, 26 p. 301 #27, 28, 30 p. 303 #1,2,3 p. 304 #6,7,8,9 (how do you know if two vectors are perpendicular? Scalar product is 0!) p.306-307 try them all TAKE HOME QUIZ DUE TOMORROW

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6.26.2 Dot Product of Vectors. What you’ll learn about How to find the Dot Product How to find the Angle Between Vectors Projecting One Vector onto Another.

6.26.2 Dot Product of Vectors. What you’ll learn about How to find the Dot Product How to find the Angle Between Vectors Projecting One Vector onto Another.

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