Vectors and the Geometry of Space

Question

Consider the line L(t)=⟨4−3t,17+4t⟩. Then L intersects:

1.The x-axis at the point.... when t =...

2.The y-axis at the point..... when t =....

3.The parabola y=x² at the points..... and... when t =....and.... .

Answer 1

L(t)=⟨4−3t,17+4t⟩.

1. Intersect x-axis: y = 0; 17 + 4t = 0; t = -17/4; x = 4 + 51/4= 67/4; (67/4, 0) is x-intercept.
2. Intersect y-axis: x = 0; 4 - 3t = 0; t = 4/3; y = 17 + 16/3 = 67/3; (0, 67/3) is y-intercept.
3. Intersection with parabola: 17 + 4t = (4 - 3t)²; 9t² - 28t - 1 = 0; t = (14 ± √205)/9;

t₁ = (14 + √205)/9, point ((- 2 - √205)/3, (167 + √205)/9))

t₂ = (14 - √205)/9, point ((-2 + √205)/3, (167 - √205)/9))