Question

derive the equation of the plane perpendicular to a given vector and passing through a given point both in vector and cartesian form

Answer 1

Answer:

Let us suppose that our varying plane vector from a fixed point is r

and, the given point through which the plane passes be a

Now, let the given vector which is perpendicular to the plane be n

Clearly, from geometrical intuition,

r-a vector will be perpendicular to the vector n

Which implies that their scalar or dot product must be zero.

So,

(r-a). n = 0

or,

r.n = a.n

This is the vector equation.

Now for the equation in Cartesian form

Suppose

r = x i^ + y j^ + z k^

and, point vector

n = n1 i^ + n2 j^ + n3 k^

Which finally leads from the vector equation that

n1 x + n2 y + n3 z = d

where, d is a scalar = a.n

Hope this helps you !!

Answer 2

Answer:

Let us suppose that our varying plane vector from a fixed point is r

and, the given point through which the plane passes be a

Now, let the given vector which is perpendicular to the plane be n

Clearly, from geometrical intuition,

r-a vector will be perpendicular to the vector n

Which implies that their scalar or dot product must be zero.

So,

(r-a). n = 0

or,

r.n = a.n

This is the vector equation.

Now for the equation in Cartesian form

Suppose

r = x i^ + y j^ + z k^

and, point vector

n = n1 i^ + n2 j^ + n3 k^

Which finally leads from the vector equation that

n1 x + n2 y + n3 z = d

where, d is a scalar = a.n