# Find Unit Vectors for f(x,y) w/ D_uf=0

• Cpt Qwark
In summary: However, that maximum value, \sqrt{2}, is irrelevant here. You are looking for two unit vectors perpendicular to \nabla f.In summary, you are asked to find two unit vectors perpendicular to the gradient of the function f(x,y)=x^2-xy+y^2. The given vector u=i+j is not relevant to this problem. The previous questions may have used u, but the maximum value of D_uf is not needed here. The directional derivative D_vf is defined as the dot product of the gradient and a unit vector in the same direction as v. Since you want D_vf to be 0, you need to find two unit vectors perpendicular to \nabla f.
Cpt Qwark

## Homework Statement

For $$f(x,y)=x^2-xy+y^2$$ and the vector $$u=i+j$$.
ii)Find two unit vectors such $$D_vf=0$$

N/A.

## The Attempt at a Solution

Not sure if relevant but the previous questions were asking for the unit vector u - which I got $$\hat{u}=\frac{1}{\sqrt{2}}(i+j)$$ for the maximum value of $$D_uf$$ which was $$\sqrt{2}$$.

The directional derivative is defined as
$$D_{v}f = \nabla f \cdot \mathbf{v}$$
Your task is to find two vectors $\mathbf{v}$ such that $D_{v}f = 0$.

Cpt Qwark said:

## Homework Statement

For $$f(x,y)=x^2-xy+y^2$$ and the vector $$u=i+j$$.
ii)Find two unit vectors such $$D_vf=0$$
This problem doesn't have anything to do with the vector "u". Why is that given? $D_vf$ is the dot product of the gradient, $\nabla f$, and a unit vector in the same direction as vector v. Since you want that to be 0, you are looking for two unit vectors perpendicular to $\nabla f$.

2. Homework Equations
N/A.

## The Attempt at a Solution

Not sure if relevant but the previous questions were asking for the unit vector u - which I got $$\hat{u}=\frac{1}{\sqrt{2}}(i+j)$$ for the maximum value of $$D_uf$$ which was $$\sqrt{2}$$.
So "u" was used in previous questions?

## 1. What are unit vectors?

Unit vectors are vectors that have a magnitude of 1 and are used to indicate direction in a given space. They are commonly used in mathematics and physics.

## 2. How do you find unit vectors for a given function?

To find unit vectors for a function, you need to take the partial derivatives of the function with respect to each variable. Then, you can use those partial derivatives to create a vector and divide it by its magnitude to get a unit vector.

## 3. What is the significance of unit vectors in vector calculus?

Unit vectors are important in vector calculus because they allow us to represent and manipulate vectors in a more simplified manner. They also help us understand the direction and magnitude of a vector in a given space.

## 4. How does finding unit vectors relate to the equation Duf=0?

The equation Duf=0 represents the directional derivative of a function in the direction of a given unit vector. By finding unit vectors for a function, we can then use them to calculate the directional derivative and determine the direction in which the function is changing the fastest.

## 5. Can unit vectors be negative?

No, unit vectors cannot be negative. As mentioned earlier, they have a magnitude of 1, which means they cannot have a negative value. However, they can indicate a direction that is opposite to the original vector's direction.

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