UNIT 8 – EQUATIONS of LINES and PLANES
LESSON 1 – VECTOR and PARAMETRIC EQUATIONS of LINES in R2
RECALL: To determine the equation of a straight line, the following is required:
a) two points on the line OR
b) one point on the line and its direction (or slope).
Example ① Determine the equation of a line that passes through the points
(-3, 0) and (3, 2).
DIRECTION VECTORS
Any vector parallel to a line may be used as a direction vector, ⃗𝒅⃗.
Example ② State a direction vector for each of the following:
a)
the line through A(1,6) and B(4,0);
b)
the line with slope 5;
_______________
c)
a horizontal line;
_______________
d)
a vertical line.
_______________
4
_______________
VECTOR EQUATION of a LINE (R2)
Given direction vector, 𝑑⃗ = (𝑎, 𝑏), and point, 𝑃𝑜 (𝑥𝑜, 𝑦𝑜 ), a line is uniquely determined through 𝑃𝑜
𝑃(𝑥, 𝑦)
and parallel to 𝑑⃗:
𝑃𝑜 (𝑥𝑜, 𝑦𝑜 )
⃗⃗⃗⃗⃗⃗⃗
𝑃𝑜 𝑃 = 𝑡𝑑⃗
⃗⃗ = (𝑎, 𝑏)
𝒅
𝑏
𝑎
Vector Equation of a Line in R2:
(𝑥, 𝑦) = (𝑥𝑜, 𝑦𝑜 ) + 𝑡(𝑎, 𝑏), 𝑡 ∈ 𝑅
OR
𝑟⃗ = (𝑥𝑜, 𝑦𝑜 ) + 𝑡(𝑎, 𝑏), 𝑡 ∈ 𝑅
Example ③ Determine the vector equation (V.E.) of the line passing through A(2,-3) and B(6,-1).
Example ④ State the V.E. of the line passing through (2,3) that is:
a) parallel to 𝑟⃗ = (0,1) + 𝑡(−4, 5);
__________________________
b) perpendicular to 𝑟⃗ = (0,1) + 𝑡(−4, 5).
__________________________
PARAMETRIC EQUATIONS of a LINE (R2)
The equation of a line, (𝑥, 𝑦) = (𝑥𝑜, 𝑦𝑜 ) + 𝑡(𝑎, 𝑏), 𝑡 ∈ 𝑅, can also be written:
𝑥 = 𝑥𝑜 + 𝑡𝑎
𝑦 = 𝑦𝑜 + 𝑡𝑏
𝑡∈𝑅
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Example ⑤ A line passes through (-2,5) with 𝑑⃗ = (− 2 , 3). Determine:
a) a direction vector with integer components;
b) the parametric equation of the line;
c) whether the point, (0,6), lies on the line;
d) the intercepts of the line;
e) the coordinates of the point where t = 3.
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