Finding the Cartesian form of the vector equation(clarified)

Question

The points on the plane are: (400, 0, 3), (-50, 500, 7) and (-200, -100, 5) So the vector equation in question is: $$ r = 400i - 3k + \lambda (-225i + 250j + 2k) + \mu(-300i - 50j + k) $$ After this, what I have done is equate $ x $ to its component, so: $$ -225\lambda - 300\mu = x - 400 $$ $$ 250\lambda - 50\mu = y $$ $$ 2\lambda + \mu = z + 3 $$

Then I used equation 3 to find that $ \mu = z + 3 -2\lambda $, then substituted that into equation 2 to get: $$ 250\lambda - 50(z + 3 -2\lambda) = y $$ $$ 250\lambda - 50z - 150 + 100\lambda = y $$ $$ 350\lambda = y + 50z + 150 $$ $$ \lambda = \frac{1}{350}(y + 50z + 150) $$

Then I substituted this into equation 1, since we haven't touched that yet when solving this set of simultaneous equations: $$ -225\lambda - 300\mu = x - 400 $$ $$ -225[\frac{1}{350}(y + 50z + 150)] - 300(z + 3 -2\lambda) = x - 400 $$ $$ -\frac{9}{14}y - \frac{225}{7}z - \frac{675}{7} - 300z - 900 + 600\lambda = x - 400 $$ $$ -\frac{9}{14}y - \frac{2325}{7}z - \frac{6975}{7} + 600[\frac{1}{350}(y + 50z + 150)] = x - 400 $$ $$ -\frac{9}{14}y - \frac{2325}{7}z - \frac{6975}{7} +\frac{12}{7}y + \frac{600}{7}z + \frac{1800}{7} = x - 400 $$ $$ -x +\frac{15}{14}y - \frac{1725}{7}z - \frac{5175}{7} = - 400 $$ $$ -x +\frac{15}{14}y - \frac{1725}{7}z = \frac{2375}{7} $$ $$ -14x + 15y - 3450z = -4750 $$ $$ 14x - 15y + 3450z = 4750 $$

This is my final answer, but the actual answer is $ 14x - 15y + 3450z = 15950 $. Can someone help me through where I went wrong or if my method is wrong, any tips and guidance and hints would be appreciated a lot as I am self-studying this.

Answer 1

Hint:

To find the Cartesian equation of a plane given its parametric form:

$$ \mathbf{r} = \mathbf{r_0} + \lambda \mathbf{d_1} + \mu \mathbf{d_2} $$

one follows these key steps:

Step 1: Find the Normal Vector of the Plane

• The normal vector $ \mathbf{n} $ to the plane is perpendicular to both direction vectors $ \mathbf{d_1} $ and $ \mathbf{d_2} $.
• Compute the cross product $ \mathbf{d_1} \times \mathbf{d_2} $ to obtain this normal vector.

Step 2: Use the Plane Equation Formula

The general equation of a plane is:

$$ n_1(x - x_0) + n_2(y - y_0) + n_3(z - z_0) = 0 $$

where:

• $ (x_0, y_0, z_0) $ is the given point on the plane ($ \mathbf{r_0} $),
• $ (n_1, n_2, n_3) $ are the components of the normal vector.

Step 3: Substitute the Values

• Replace $ (x_0, y_0, z_0) $ with the coordinates of $ \mathbf{r_0} = (400, 0, -3) $.
• Use the computed normal vector $ (n_1, n_2, n_3) $ to form the equation.

Step 4: Expand and Simplify

• Expand the equation to get it in standard Cartesian form:
  $$ Ax + By + Cz + D = 0 $$
• Where $ A, B, C $ come from the normal vector and $ D $ is obtained from substituting $ (x_0, y_0, z_0) $.

Details are left to you. I hope it helps.

Answer 2

I think important to know that such a problem can be easily solved with an algebraic method coping only with points, i.e., without needing to transform the issue into a vector issue (even if the vector approach is highly valuable because it gives a geometrical understanding of the question).

Here is the method, based on a (classical) co-planarity criteria for 4 points $A,B,C,D$ is as follows :

$$\det\pmatrix{x_A&y_A&z_A&\color{red}{1}\\x_B&y_B&z_C&\color{red}{1}\\x_C&y_C&z_C&\color{red}{1}\\x_D&y_D&z_D&\color{red}{1}}=0$$

In our issue, we have three given points $A,B,C$ ; the fourth point $D$ is the generic point $(x,y,z)$ of the looked-for plane. Therefore, its equation is :

$$\det\pmatrix{400&0&3&\color{red}{1}\\-50&500&7&\color{red}{1}\\-200&-100&5&\color{red}{1}\\x&y&z&\color{red}{1}}=0$$

It remains to expand it, for example with a Laplace extension wrt the last line by hand (which means the computation four $3 \times 3$ determinants).

An alternative is to let a Computer Algebra System do this kind of tedious (and prone to errors) work for you.

See below a SAGE program for this purpose that you can run easily ; SAGE is an easily usable public domain software : just type https://sagecell.sagemath.org/ ; an edit window opens ; copy-paste this program into this window, then press button "Evaluate" :

 var('x y z')
 M=matrix([[400, 0, 3,1],
           [-50, 500, 7,1],
           [-200, -100, 5,1],
           [x,y,z,1]])
 d=M.determinant()
 print(d)
result : -1400x + 1500y - 345000*z + 1595000

(please note the unimportant $-100$ factor)