1. A line l in R^3 is given by r(t) = <1-t, 2t, 2+2t>, t ? R. How far is this line from the poin

1. First, we need to find the direction vector of the line. From the given equation, we have the

Question

1. A line l in R^3 is given by r(t) = <1-t, 2t, 2+2t>, t ? R. How far is this line from the point P_0 = (1, 1, 2)? 2. (a) u = <-4, 0, 2>, v = <0, 1, 1> are two given vectors. Compute Comp_v^u (b) Suppose that v, in part (a), is a direction vector for a line, which also contains the point (1, 2, 3). Determine the symmetric equation of this line. 3. Determine the equation of the plane containing the non-collinear points P(1, 2, 1), Q(-1, 1, 2) and R(1, 1, 1). 4. Suppose that r(t) is the position vector of some motion in 3-space, such that at the instant t_0, r(t_0) = <2, 1, -1>, T = <1/?5, 2/?5, 0>, N = <-2/?5, 1/?5, 1>. Determine the equation of the osculating plane. 5. A vector-valued function is given by r(t) = <e^-t sin t, e^-t cost, e^-t>, t ? R. Compute T, N and B at the instant t=0. 6. (a) Compute the integral ? <cos 2t, t/(1+t^2)> dt. (b) Are the points A(1, 1, 2), B(-1, -2, 1) and C(0, 0, 4) collinear? Justify your answer.

          1. A line l in R^3 is given by r(t) = <1-t, 2t, 2+2t>, t ? R. How far is this line from the point P_0 = (1, 1, 2)?
2. (a) u = <-4, 0, 2>, v = <0, 1, 1> are two given vectors. Compute Comp_v^u
(b) Suppose that v, in part (a), is a direction vector for a line, which also contains the point (1, 2, 3). Determine the symmetric equation of this line.
3. Determine the equation of the plane containing the non-collinear points P(1, 2, 1), Q(-1, 1, 2) and R(1, 1, 1).
4. Suppose that r(t) is the position vector of some motion in 3-space, such that at the instant t_0, r(t_0) = <2, 1, -1>, T = <1/?5, 2/?5, 0>, N = <-2/?5, 1/?5, 1>. Determine the equation of the osculating plane.
5. A vector-valued function is given by r(t) = <e^-t sin t, e^-t cost, e^-t>, t ? R. Compute T, N and B at the instant t=0.
6. (a) Compute the integral ? <cos 2t, t/(1+t^2)> dt.
(b) Are the points A(1, 1, 2), B(-1, -2, 1) and C(0, 0, 4) collinear? Justify your answer.

1. A line l in R^3 is given by r(t) = <1-t, 2t, 2+2t>, t ? R. How far is this line from the point P0 = (1, 1, 2)?
2. (a) u = <-4, 0, 2>, v = <0, 1, 1> are two given vectors. Compute Compv^u
(b) Suppose that v, in part (a), is a direction vector for a line, which also contains the point (1, 2, 3). Determine the symmetric equation of this line.
3. Determine the equation of the plane containing the non-collinear points P(1, 2, 1), Q(-1, 1, 2) and R(1, 1, 1).
4. Suppose that r(t) is the position vector of some motion in 3-space, such that at the instant t0, r(t0) = <2, 1, -1>, T = <1/?5, 2/?5, 0>, N = <-2/?5, 1/?5, 1>. Determine the equation of the osculating plane.
5. A vector-valued function is given by r(t) = <e^-t sin t, e^-t cost, e^-t>, t ? R. Compute T, N and B at the instant t=0.
6. (a) Compute the integral ? <cos 2t, t/(1+t^2)> dt.
(b) Are the points A(1, 1, 2), B(-1, -2, 1) and C(0, 0, 4) collinear? Justify your answer.

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