1. 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.

2. 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

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.
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

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.
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

5. 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

6. 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, 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

7. 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, 𝑎 = −3, 8
𝑥+4,𝑦−11 =𝑡 −3, 8
𝑃 1, 𝑎 = −7, 2
𝑥− 𝑥 1 ,𝑦− 𝑦 1 =𝑡 𝑎 𝑥 , 𝑎 𝑦
𝑥−1,𝑦−5 =𝑡 −7, 2
𝑃 −5, 𝑎 = 4, −3
𝑥+5,𝑦−2 =𝑡 4, −3

8. 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

9. 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

10. 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−
𝑦=−2 𝑥− 𝑦=− 2 3 𝑥 𝑦=− 2 3 𝑥+ 14 3

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=−7𝑡+1 𝑡= 𝑥−1 −7
𝑦=2𝑡+5
𝑦=2 𝑥−1 −7 +5
𝑦=− 2 7 𝑥 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

12. 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
𝑦=− 3 4 𝑥− 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− 𝑦=− 8 3 𝑥+ 1 3

13. 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 .

14. 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 3 𝑥−2 +9=− 2 3 𝑥 =− 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

15. 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.