Find an equation of the plane through the point and perpendicular to the given vector.
(7, 1, 5)
‹-3, 1, 2›
I think I need the normal vector, but how do I find it? And then what do I do?
Thank you.
The equation of this plane is
-3(x-7)+1(y-1)+2(z-5)=0 or
-3x+y+2z+10=0
To find an equation of the plane passing through a given point and perpendicular to a given vector, we first need to find the normal vector of the plane. This can be done using the given vector.
The normal vector of a plane is always perpendicular to the plane. Therefore, it will be orthogonal (perpendicular) to the given vector. To find the normal vector, we can use the dot product.
The dot product of two vectors, v and w, is given by the formula: v ⋅ w = |v| |w| cosθ, where θ is the angle between the two vectors.
In this case, we have the given vector ‹-3, 1, 2› and we want to find a normal vector. We can choose any vector that is orthogonal to ‹-3, 1, 2›.
Let's assume the normal vector is ‹a, b, c›. Since the normal vector is perpendicular to the given vector, the dot product of the two vectors should be zero:
‹-3, 1, 2› ⋅ ‹a, b, c› = 0
Using the dot product formula, we can expand this equation:
(-3a) + (b) + (2c) = 0
Now we have an equation with three variables (a, b, c) and one equation. To find a specific solution, we need another equation. Since we are given a point (7, 1, 5) that lies on the plane, we can substitute these values into the equation of the plane:
(-3a) + (b) + (2c) = 0
Substitute (7, 1, 5) into the equation:
(-3a) + (1) + (2c) = 0
Simplifying this equation will give us a system of two equations with two variables (a and c). We can then find the values of a, b, and c.
Once you have the values of a, b, and c, you can write the equation of the plane in the form ax + by + cz = d, where (x, y, z) represents any point on the plane. The values of a, b, c will be the coefficients of x, y, z respectively, and d will be a constant term.
I hope this explanation helps you in finding the equation of the plane passing through the given point and perpendicular to the given vector.