Is the concept of the same direction in vector different from that in plane geometry?

The parallel problem in vector is different from that in geometry
In vectors, parallel is collinear, collinear is parallel, and vectors with the same direction are collinear and parallel
In geometry, parallel and coincidence of lines are two kinds of positional relations

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1. Space geometric vector It is known that the radius of the circumscribed sphere o of the triangular pyramid P-A B C is 1 and satisfies the vector OA + ob + OC = 0 Then the volume of the regular triangular pyramid P-A B C?
2. Find the normal vector (urgent)! Known surface equation is x ^ 2 + 2Y ^ 2 + 3Z ^ 2 = 15, find the normal vector
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5. Vector computing Vector contains direction and size, but with direction operation does not change the size of it?
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7. How to calculate the length of a complex vector? Suppose I have a two-dimensional complex vector a, how can I prove that the square of its length is equal to the conjugate product of a and a? In other words, why is it not equal to the product of a and a itself? I know that the conjugate of a and a can guarantee that the product is positive, but why is the length calculated in this way?
8. Why can vectors (vectors in Physics) be calculated from geometry? How can we prove that everything obeys this law? Why should we abide by this Law and not others?
9. How to calculate the length of a vector
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12. How to find normal vector and direction vector? For example, 5x + 6y-1 = 0, how can we easily see the normal vector and direction vector of this equation?
13. What does the normal vector of a plane mean?
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20. Is work scalar or vector?