For the following questions, say you were given a line and a plane as below ...

For the following questions, say you were given a line and a plane as below

Line: r(t)= < x(t), y(t), z(t) > = < t-2, t+1, 3t > , Plane: : a(x+2) + b(y-3) - 4z = 2

a) What relationship would have to exist between scalars a and b for the line not to intersect with the plane?

Hint 1) Plug in x, y, z given in the line into the plane equation.

Hint 2) Say you were solving this equation and got the following. What would that indicate?

i) t = 2: ________________________________________________________

ii) 6 = –5: ________________________________________________________

iii) 4 = 4: ________________________________________________________

So what must be true about the coefficient of t? What also can’t be true about the constant term?

b) What relationship would have to exist between scalars a and b for the plane to contain the line?

c) Assuming the line is not contained in the plane but does intersect it, give an expression for the time t that the line intersects the plane. Also give the point of intersection.
(Both answers will have scalars a and b in them)

Expert Solution

Colby Messinger answered 5 months ago

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