Question

Find the vector equation of the plane passing through the points (0,0,0),(0,5,0).and (2,0,1)
​

Answer 1

the vector equation of the plane passing through the points (0,0,0),(0,5,0).and (2,0,1) is 5x - 10z

The vector equation of the plane passing through the points (0,0,0),(0,5,0).and (2,0,1) is given by equation,

$\left[\begin{array}{ccc} x-x1 &y-y1&z-z1\\ x2-x1 &y2-y1&z2-z1\\ x3-x1 &y3-y1&z3-z1\end{array}\right]$

Given,

(0,0,0) = a = 0i + 0j + 0k ⇒ x1 = 0, y1 = 0, z1 = 0

(0,5,0) = b = 0i + 5j + 0k ⇒ x2 = 0, y2 = 5, z2 = 0

(2,0,1) = c = 2i + 0j + 1k ⇒ x3 = 2, y3 = 0, z3 = 1

Now, we have,

$\left[\begin{array}{ccc} x-0 &y-0&z-0\\ 0-0 &5-0&0-0\\ 2-0 &0-0&1-0\end{array}\right]$

$\left[\begin{array}{ccc} x &y&z\\ 0 &5&0\\ 2 &0&1\end{array}\right]$

= x(5-0) - y(0-0) + z(0-10)

= 5x - 10z

Hence the required equation.