Given 2x -3y + z - 6 = 0, how do you get a vector equation from this scalar or a parametric equations?

Question

Evelyn Agard · Accepted Answer

#pi: vec r = ((0),(0),(6)) + s ((1),(0),(-2)) + t ((0),(1),(3)) #

if

#vec n# is the normal vector to the plane #pi#,

#r_o# is a specific point lying on #pi#,

#vec r# is any other point on #pi#

then #(\vec r - vec r_o) # is a line lying along #pi#

....and so we can say using the scalar dot product that

#(\vec r - vec r_o) * vec n = 0#, which is the vector equation of the plane #pi#

we can re-arrange that to #\vec r * vec n - vec r_o * vec n = 0#

and, in Cartesian with #vec r = ((x),(y),(z))#, we can match it up to your plane as follows using the scalar dot product

#((x),(y),(z)) * ((2),(-3),(1)) - 6 = 0#

for a parametric form, we will need a point #vec p_o# on plane #pi#, and 2 non-parallel vectors, #vec u# and #vec v#, that run along #pi#. We can then identify any point on #pi# as

#vec r = vec p_0 + s vec u + t vec v# where s and t are the paremeters.

generating a #vec p_o # is simple. we just take #2x -3y + z - 6 = 0# and set #x = y = 0# so that #vec p_o = ((0),(0),(6))#

next we want #vec u# and # vec v# to be orthogonal to #vec n#

Again using the scalar dot product, that means #vec u * vec n = vec v * vec n = 0#

So for #vec u# we have #2 u_1 - 3 u_2 + u_3 = 0#

We can just arbitrarily choose #u_1 = 1, u_2 = 0 implies u_3 = -2# giving #vec u = ((1),(0),(-2))#

For #vec v# we have #2 v_1 - 3 v_2 + v_3 = 0# but we must ensure that #vec v# is not parallel to #vec u#

So this time we can just arbitrarily choose #v_1 = 0, v_2 = 1 implies v_3 = 3# giving #vec v = ((0),(1),(3))#

Now we can check our work!! We would expect the result #vec u times vec v = vec n# as #vec u# and #vec v# lie in #pi# and #vec n# is it's normal vector

#vec u times vec v = det ((hat x, hat y, hat z),(1,0,-2),(0,1,3))#

#=2 hat x - 3 hat y + 1 hat z# BINGO!!!!

one form of parameterised plane #pi# is therefore

#pi: vec r = ((0),(0),(6)) + s ((1),(0),(-2)) + t ((0),(1),(3)) #

Luke Alvarez · Answer

undefined To obtain a vector equation or parametric equations from the given scalar equation $2x - 3y + z - 6 = 0$, follow these steps:

1. Solve the equation for one of the variables (usually $z$) in terms of the other variables.
2. Express the other variables (usually $x$ and $y$) in terms of parameters.
3. Write the equations in vector or parametric form.

Here's how to do it:

1. Solve the equation for $z$:
$$z = 6 - 2x + 3y$$

2. Choose parameters for $x$ and $y$. Let's use $s$ and $t$, respectively.

3. Express $x$ and $y$ in terms of $s$ and $t$:
$$x = s$$
$$y = t$$

4. Substitute these expressions into the equation for $z$:
$$z = 6 - 2s + 3t$$

5. Write the equations in vector form:
$$\mathbf{r} = \begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} s \ t \ 6 - 2s + 3t \end{pmatrix}$$

6. Write the parametric equations:
$$x = s$$
$$y = t$$
$$z = 6 - 2s + 3t$$

So, the vector equation is $\mathbf{r} = \begin{pmatrix} s \ t \ 6 - 2s + 3t \end{pmatrix}$ and the parametric equations are $x = s$, $y = t$, and $z = 6 - 2s + 3t$.