How to find the normal vector of a plane with determinant

How to find the normal vector of a plane with determinant

In general, it is not necessary to use determinant, but write the normal vector directly;
For example, the normal vector of 3x-5t + 4z-7 = 0 is {3, - 5,4} = 3i-5j + 4K
But if you know two vectors (not parallel) or three points on the plane
(not collinear), a normal vector can be represented by a determinant
① α = {a, B, C}, β = {D, e, f} are two vectors on the plane
The normal vector can be expressed as α × β = determinant
|i j k|
|a b c|
|D E F | denotes
② A (A1, B1, C1), B (A2, B2. C2). C (A3, B3, C3) is a plane
The top three points (not collinear),
Then the normal vector can be expressed as ab × BC = determinant
| i,j,k.|
|a2-a1,b2-b1,c2-c1|
|A3-a2, b3-b2, c3-c2 | denote

Why can space vector cross product be written as a third-order determinant, and plane vector need not be multiplied by unit vector

Well, strictly speaking, the cross product of vector is a third-order determinant. Plane vector is written as a second-order determinant for convenience because it lacks the z-direction component (in fact, it should be written in the form of (x, y, 0)). Normally, plane vector (x1, y1,0) * (x2, y2,0) should be written as the following determinant:
i j k
x1 y1 0
x2 y2 0
Because in the calculation of I direction and j direction components, the result is always 0, so only K direction has components, which is reasonable. The second-order form only simplifies the calculation, which is not standard. It is best not to use it in formal occasions

What is the law of vector derivation?

In the linear coordinate system, each component of the space vector is a three variable function that changes with the position. The derivation of the vector can be converted into the derivation of the function. For details, see the textbook of mathematical analysis. There is a chapter dedicated to the derivation of the vector
There is an e-book "mathematical analysis course" on the Internet, Volume 1 of song Guozhu, which is very eye-catching

Vector product formula
Let a = I + 2j-k, B = 2J + 3k, then what is the vector product of a and B

The vector product of a and B is: 8i-3j + 2K