Question

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?

Answer 1

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`