How do you find a vector parametric equation r(t) for the line through points P=(-3,-1,1) and Q = (-8,-4,5) If r(6) = P and r(10) = Q?

1 Answer
Nov 15, 2016

Please see the explanation.

Explanation:

The general form of the 3D vector equation of a line is a point plus a vector multiplied by a scalar, t:

#r(t) = (x_p, y_p, z_p) + t(x_v,y_v,z_v)#

The parametric equations are:

#x = tx_v + x_p#
#y = ty_v + y_p#
#z = tz_v + z_p#

Equations [1] and [2] are the x parametric equation evaluated at 6 and 10 respectively:

#-3 = 6x_v + x_p" [1]"#
#-8 = 10x_v + x_p" [2]"#

Eliminate #x_p# by subtracting equation [1] from equation [2]

#-5 = 4x_v#

Solve for #x_v#:

#x_v = -5/4#

Substitute #-5/4# for #x_v# in the general equation:

#r(t) = (x_p, y_p, z_p) + t(-5/4,y_v,z_v)#

Substitute for #x_v# in equation [1]:

#6(-5/4) + x_p = -3#

Solve for #x_p#:

#x_p = 9/2#

Substitute #9/2# for #x_p# in the general equation:

#r(t) = (9/2, y_p, z_p) + t(-5/4,y_v,z_v)#

Equations [3] and [4] are the y parametric equation evaluated at 6 and 10 respectively:

#-1 = 6y_v + y_p" [3]"#
#-4 = 10y_v + y_p" [4]"#

Subtract [3] from [4]:

#-3 = 4y_v#

Solve for #y_v#

#y_v = -3/4#

Substitute into the general equation:

#r(t) = (9/2, y_p, z_p) + t(-5/4,-3/4,z_v)#

Substitute #-3/4# for #y_v# in equation [3]:

#-1 = 6(-3/4) + y_p#

Solve for #y_p#:

#y_p = 7/2#

Substitute into the general equation:

#r(t) = (9/2, 7/2, z_p) + t(-5/4,-3/4,z_v)#

Equations [5] and [6] are the z parametric equation evaluated at 6 and 10 respectively:

#1 = 6z_v + z_p" [5]"#
#5 = 10z_v + z_p" [6]"#

Subtract equation [5] from equation [6]:

#4 = 4z_v#

Solve for #z_v#

#z_v = 1#

Substitute into the general equation:

#r(t) = (9/2, 7/2, z_p) + t(-5/4,-3/4,1)#

Substitute 1 for #z_v# in equation [5]:

#1 = 6 + z_p#

#z_p = -5#

Substitute into the general equation:

#r(t) = (9/2, 7/2, -5) + t(-5/4,-3/4,1)#

The above is the vector equation.

The parametric equations are:

#x = -5/4t + 9/2#

#y = -3/4t + 7/2#

#z = t - 5#

The symmetric equations are:

#(x - 9/2)/(-5/4) = (y - 7/2)/(-3/4) = (z - 5)/1#