Vector and parametric equation

Question

1) What is the vector and parametric equation for:

the horizontal line through (pi,e)

2) At time t, the position of an object moving with constant velocity is given by the parametric equations x=2-3t and y=-1+2t

a) When and where does it cross the line x+y=2

Answer 1

Using bold for vectors:

The vector equation for a line looks like:

R(t) = R₀ + tV where R₀ is the postion vector of a point on the line, V is a vector parallel to the line, and t is a paremeter.

1) (pi,e) is the position vector of a point on the line. (1.0) is a horizontal vector (parallel to line); so

R(t) = (pi,e) + t(1,0)

The parametric form is the same information written a different way.

R(t) = (x,y) = (pi,e) + t(1,0)

Write the x and y parts as separate equations.

x = pi + 1*t x = pi + t

y = e + 0*t y = e

Answer 2

#2:

Find t for the crossing by simply plugging in the parametric equations to the equation of the line:

x + y = 2

(2 - 3t) + (-1 + 2t) = 2

1 - t = 2

t = -1

finally, solve the 2 parametric equations at t = -1

#1:

Parametric equation:

y = e

If you are given the definition of a straight line with no limits, I don't know how that could be described as a vector.