b Find the equation of the line perpendicular to the one in a and passing

B find the equation of the line perpendicular to the

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( b): Find the equation of the line perpendicular to the one in (a), and passing through Q(1,4). Solution: We need vector & such that & . ° = 0, with ° = 1 2 . ()* & + & , . 1 2 = 0 or & + + 2& , = 0. Let & , = −1, then & + = 2 , or & = 2 −1 . Then - = . / = ¹ + 0& = 1 4 + 0 2 −1 . 1
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9/2/2015 4 ( c): Find the coordinates of the point of intersection between the two lines. Solution: We need point R. 2±¼3 ¸¿ = (., /) = 2,2 + 4(1,2) 2±¼3¹¿ = ., / = 1,4 + 0(2, −1) Now the x-values and y-values are the same at intersection: For BR: . = 2 + 4 / = 2 + 2 4 For QR: . = 1 + 2 0 / = 4 − 1 0 2 + 4 = 1 + 20 5±637 4 = −1 + 20 And 2 + 24 = 4 − 10 (1) Substitute the 4 value in (1): 2 + 2 −1 + 20 = 4 − 0 2 − 2 + 40 = 4 − 0 50 = 4 0 = 8 9 = 0.8 But 4 = −1 + 20 4 = −1 + 2 0.8 4 = 0.6 From: For BR: . = 2 + 4 / = 2 + 2 4 . = 2 + 0.6 = 2.6 °±; / = 2 + 2 0.6 = 3.2 For QR: . = 1 + 2 0 / = 4 − 1 0 . = 1 + 2 0.8 = 2.6 °±; / = 4 − 0.8 = 3.2 So: ² = (2.6, 3.2)
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9/2/2015 5 EXAMPLE Example 3 . Find the line of intersection between the planes . + / + < = 2 and 2. − < = 0 . Solution: Parameterize (Create a parameter): Choose . = 4, (you can choose any of x, y or z) Then . = 4 < = 2. = 24 / = 2 − . − < = 2 − 4 − 24 = 2 − 34 . / < = 4 2 − 34 24 Take apart: . / < = 4 2 − 34 24 = 0 + 4 2 − 34 0 + 24 = 4 −34 24 + 0 2 0 = 0 2 0 + 4 1 −3 2 , => - = 0 2 0 + 4 1 −3 2 Vector equation of the line of intersection.
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9/2/2015 6 NOW YOU: Find the vector equation of the line of intersection between the planes: 2 . − / + < = 2, 2. + / + 3< = 6. Next find the angle between the planes. ANOTHER ONE FOR YOU TO TRY… Find the scalar equation of the plane which contains 3 points P 0 , P 1 and P 2 at any position on the plane. Scalar equation: °. + ?/ + &< = ; @ A = 4,5, −7 @ B = 1,2, −3 @ ± = (0,1,0) C D C ² C E
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9/2/2015 7 MORE EXAMPLES We can not handle all the examples in class with all the different combinations of lines and planes. Make sure to go through ALL the additional exercises that are on Blackboard in connection with the lines and planes. If you should have any problems, make sure to attend the practical sessions on Tuesday (WWG113) or Wednesday (W201) to ask for assistance OR attend the walk-in hours. It is YOUR responsibility to get enough exercise with this.
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