Mathgician said:dot product, or cosine of theta
positive if theta is 0, negative if theta is 180 or pi
that tells your max and min slope
oahsen said:what should ı do with the max and min slope value?
Mathgician said:what is the vector from f(0,0,a) to f(0,0,-a)?
find the unit vector of this vector,
and then do the dot product of the unit vector and the gradient vector.
oahsen said:since we do not know f how can we find the vector from f(0,0,a) to f(0,0,-a)?
please be more clear. I do not know perfectly this subject.
Mathgician said:lets call f(0,0,a) point P, and f(0,0,-a) point S
What is vector PS?
When the gradient is parallel to the position vector, it means that the rate of change of a function in a particular direction (represented by the gradient) is equal to the direction of movement (represented by the position vector).
To determine if the gradient is parallel to the position vector, you can calculate the dot product of the gradient vector and the position vector. If the dot product is equal to zero, then the vectors are perpendicular and the gradient is not parallel to the position vector.
The significance of the gradient being parallel to the position vector is that it indicates a direction of maximum change for a given function. This can be useful in optimization problems, where finding the direction of maximum change can help determine the most efficient solution.
Yes, the gradient and position vector can be parallel in certain cases. This occurs when the function is constant in a particular direction, meaning there is no change in that direction and the gradient is zero. In this case, any direction can be considered parallel to the position vector.
The concept of parallel gradient and position vector is an important aspect of vector calculus, as it allows for the calculation of directional derivatives and optimization in multiple variables. It also helps to understand the relationship between the gradient and level curves/surfaces of a function.