# Question Video: Finding the Equation of a Plane Perpendicular to a Given Vector and Passing through a Given Point Mathematics

Find the equation of the plane which is perpendicular to the vector π = 5π’ β 7π£ β 3π€ and passes through the point π΅(β5, 5, 9).

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### Video Transcript

Find the equation of the plane which is perpendicular to the vector π equals five π’ minus seven π£ minus three π€ and passes through the point π΅ negative five, five, nine.

Okay, here weβre trying to find the equation of a plane. And letβs say that this is that plane. And weβre told that this vector π is perpendicular to our plane. That means if we sketched in vector π, it would look something like this. We also know that our plane passes through the point called point π΅. What we have then is a vector that is normal or perpendicular to our plane and a point the plane passes through. This is actually all we need to determine the equation of our plane.

Letβs recall the vector form of a planeβs equation. This form tells us that the dot product of a vector normal to the plane and a vector to a general point in it is equal to the dot product of that normal vector and a vector to a known point in the plane. In our case, the vector π is our normal vector. And if we were to set up a coordinate frame, say, here, then a vector from the origin of that frame to point π΅ we could call π« zero. Thatβs our vector to a known point in the plane. π« zero then has components that are equal to the coordinates of point π΅. And since our normal vector π§ is identical to the vector π, it has components five, negative seven, negative three.

We can now use these values in our vector form to solve for the equation of our plane. The normal vector dotted with a vector to a general point in our plane looks like this and that normal vector dotted with a vector to point π΅ looks like this. Carrying out these dot products, we get five π₯ minus seven π¦ minus three π§ equals negative 25 minus 35 minus 27, which adds up to negative 87. If we now add positive 87 to both sides of this equation, then we find that five π₯ minus seven π¦ minus three π§ plus 87 equals zero. This is what is called the general form of the equation of our plane.