8.6 Vector and Parametric equations By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity.

Equations of lines Vector Parametric Point Slope By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity. Equations of lines Distribute the 𝑡 Vector → Parametric Set component = and solve for x, y Vector 𝑥− 𝑥 1 ,𝑦− 𝑦 1 =𝑡 𝑎 𝑥 , 𝑎 𝑦 Parametric 𝑥=𝑡 𝑎 𝑥 + 𝑥 1 𝑦=𝑡 𝑎 𝑦 + 𝑦 1 Point Slope 𝑦− 𝑦 1 =𝑚 𝑥− 𝑥 1 Cartesian (Slope intercept) 𝑦=𝑚𝑥+𝑏 Parametric → Point Slope Solve 𝑥 equation for 𝑡, substitute into 𝑦 equation Point Slope → Slope Intercept simplify for 𝑦

By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity. Graphing On the left side of the handout, draw a vector of your choosing Plot a point somewhere inside the shaded box (to ensure enough space) Define your point and vector. Starting from the point, draw several copies (positive and negative) of your vector. What did you just draw? 𝟐 𝒂 𝒂 − 𝒂 −𝟐 𝒂 1 3 1/3 2 1

By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity. Graphing On the right side of the handout, complete the table using the given values of 𝑡 For the value of 𝑡=0 graph an ×, for every other value, graph a point. 𝑦=−3 −1 −4=−1 −8, −1 𝑥=6 −1 −2=−8 𝑥=6 0 −2=−2 𝑦=−3 0 −4=−4 −2, −4 4, −7 𝑥=6 1 −2=4 𝑦=−3 1 −4=−7 − 𝒂 𝒂

JUST Watch and be amazed By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity. JUST Watch and be amazed 𝑥=𝑡 𝑎 𝑥 + 𝑥 1 𝑥− 𝑥 1 =𝑡 𝑎 𝑥 𝑥− 𝑥 1 𝑎 𝑥 =𝑡 𝑦=𝑡 𝑎 𝑦 + 𝑦 1 𝑦= 𝑥− 𝑥 1 𝑎 𝑥 𝑎 𝑦 + 𝑦 1 𝑦= 𝑎 𝑦 𝑎 𝑥 𝑥− 𝑥 1 + 𝑦 1 𝑦− 𝑦 1 = 𝑎 𝑦 𝑎 𝑥 𝑥− 𝑥 1 OMG Ms. Munch…that is the SLOPE That’s our initial y value That’s our initial x-value

The 𝑡 is how many TIMES the vector is repeated By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity. Example 1: Write the vector equation of the line through 𝑃 1 𝑥 1 , 𝑦 1 parallel to 𝑎 = 𝑎 𝑥 , 𝑎 𝑦 The 𝑡 is how many TIMES the vector is repeated 𝑃 1, 4 𝑎 = 3, −2 𝑥− 𝑥 1 ,𝑦− 𝑦 1 =𝑡 𝑎 𝑥 , 𝑎 𝑦 𝑥−1,𝑦−4 =𝑡 3, −2 This is the VECTOR equation of a line. You need to know the difference between the forms

By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity. Example 1: Write the vector equation of the line through 𝑃 1 𝑥 1 , 𝑦 1 parallel to 𝑎 = 𝑎 𝑥 , 𝑎 𝑦 𝑃 3, −9 𝑎 = −1, 2 𝑥− 𝑥 1 ,𝑦− 𝑦 1 =𝑡 𝑎 𝑥 , 𝑎 𝑦 𝑥−3,𝑦+9 =𝑡 −1, 2 𝑃 −4, 11 𝑎 = −3, 8 𝑥+4,𝑦−11 =𝑡 −3, 8 𝑃 1, 5 𝑎 = −7, 2 𝑥− 𝑥 1 ,𝑦− 𝑦 1 =𝑡 𝑎 𝑥 , 𝑎 𝑦 𝑥−1,𝑦−5 =𝑡 −7, 2 𝑃 −5, 2 𝑎 = 4, −3 𝑥+5,𝑦−2 =𝑡 4, −3

Set component = and solve for x, y By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity. Example 2: Given the vector form of an equation, write the parametric form. Distribute the 𝑡 Vector → Parametric Set component = and solve for x, y 𝑥−1,𝑦−4 =𝑡 3, −2 𝑥−1,𝑦−4 = 3𝑡, −2𝑡 𝑥−1=3𝑡 𝑥=3𝑡+1 𝑦−4=−2𝑡 𝑦=−2𝑡+4 Parametric Form 𝑥=3𝑡+1 𝑦=−2𝑡+4

By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity. Example 2: Given the vector form of an equation, write the parametric form. 𝑥−1,𝑦−5 =𝑡 −7, 2 𝑥−1=−7𝑡 𝑥=−7𝑡+1 𝑦−5=2𝑡 𝑦=2𝑡+5 𝑥+5,𝑦−2 =𝑡 4, −3 𝑥+5=4𝑡 𝑥=4𝑡−5 𝑦−2=−3𝑡 𝑦=−3𝑡+2 𝑥−3,𝑦+9 =𝑡 −1, 2 𝑥−3=−𝑡 𝑥=−𝑡+3 𝑦+9=2𝑡 𝑦=2𝑡−9 𝑥+4,𝑦−11 =𝑡 −3, 8 𝑥+4=−3𝑡 𝑥=−3𝑡−4 𝑦−11=8𝑡 𝑦=8𝑡+11

Example 3: Given parametric form, convert to Cartesian form By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity. Example 3: Given parametric form, convert to Cartesian form 1. Solve for 𝑡 in the 𝑥 equation 2. Plug 𝑡 into 𝑦 equation 3. Simplify 𝑥=3𝑡+1 𝑥−1=3𝑡 𝑡= 𝑥−1 3 𝑦=−2𝑡+4 𝑦=−2 𝑥−1 3 +4 𝑦=−2 𝑥−1 3 +4 𝑦=− 2 3 𝑥+ 2 3 + 12 3 𝑦=− 2 3 𝑥+ 14 3

Example 3: Given parametric form, convert to Cartesian form By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity. Example 3: Given parametric form, convert to Cartesian form 1. Solve for 𝑡 in the 𝑥 equation 2. Plug 𝑡 into 𝑦 equation 3. Simplify 𝑥=−7𝑡+1 𝑡= 𝑥−1 −7 𝑦=2𝑡+5 𝑦=2 𝑥−1 −7 +5 𝑦=− 2 7 𝑥+ 2 7 + 35 7 𝑦=− 2 7 𝑥+ 37 7 𝑥=−𝑡+3 𝑡=−𝑥+3 𝑦=2𝑡−9 𝑦=2 −𝑥+3 −9 𝑦=−2𝑥+6−9 𝑦=−2𝑥−3

Example 3: Given parametric form, convert to Cartesian form By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity. Example 3: Given parametric form, convert to Cartesian form 1. Solve for 𝑡 in the 𝑥 equation 2. Plug 𝑡 into 𝑦 equation 3. Simplify 𝑥=4𝑡−5 𝑡= 𝑥+5 4 𝑦=−3𝑡+2 𝑦=−3 𝑥+5 4 +2 𝑦=− 3 4 𝑥− 15 4 + 8 4 𝑦=− 3 4 𝑥− 7 4 𝑥=−3𝑡−4 𝑡= 𝑥+4 −3 𝑦=8𝑡+11 𝑦=8 𝑥+4 −3 +11 𝑦=− 8 3 𝑥− 32 3 + 33 3 𝑦=− 8 3 𝑥+ 1 3

By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity. Summary Write an equation in slope intercept form of the line with parametric equations 𝑥=9𝑡+2, 𝑦=−6𝑡+9. Write a vector and parametric equation of the line that passes through the point 𝑃 −1, 6 and is parallel to 𝑎 = 3, −1 .

By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity. Summary Write an equation in slope intercept form of the line with parametric equations 𝑥=9𝑡+2, 𝑦=−6𝑡+9. 𝑥=9𝑡+2→𝑡= 𝑥−2 9 𝑦=−6 𝑥−2 9 +9=− 2 3 𝑥−2 +9=− 2 3 𝑥+ 4 3 + 27 3 =− 2 3 𝑥+ 31 3 Write a vector and parametric equation of the line that passes through the point 𝑃 −1, 6 and is parallel to 𝑎 = 3, −1 . Vector: 𝑥+1, 𝑦−6 =𝑡 3, −1 𝑥+1=3𝑡→𝑥=3𝑡−1 𝑦−6=−𝑡→𝑦=−𝑡+6 Parametric: 𝑥=3𝑡−1 𝑦=−𝑡+6

By the end of the section students will be able to write the vector equations of a line, write a parametric form of a line, and write slope intercept form of a line as evidenced by a sorting activity.