Finding the scalar equation of a plane

Formula for the scalar equation of a plane

To find the scalar equation of a plane, we’ll use the formula

$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$

where $P_0(x_0,y_0,z_0)$ is a given point and $v=\langle a,b,c\rangle$ is the normal vector to the plane. The vector may also be in the format $v=ai+bj+ck$.

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How to find the scalar equation of a plane

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Scalar equation for a plane, given a vector and a point

Example

Find the scalar equation of the plane.

$P(1,4,-8)$

$\langle3,6,2\rangle$

Plugging the given point and the given vector into our formula, we get

$3(x-1)+6(y-4)+2\left[z-(-8)\right]=0$

$3(x-1)+6(y-4)+2(z+8)=0$

$3x-3+6y-24+2z+16=0$

$3x+6y+2z=11$

The scalar equation of the plane is given by $3x+6y+2z=11$.

IF we have a vector and a point, we can find the scalar equation of a plane.

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