Multi-Variable Calculus: Cross Product Expressions

In summary, the conversation discusses using dot product and cross product notation to describe various vector operations. These include finding a vector orthogonal to two cross products, finding a vector orthogonal to two vector additions and subtractions, finding a vector with a specific length in the direction of another vector, and finding the area of a parallelogram determined by two vectors.
  • #1
Dembadon
Gold Member
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I would like to check my answers...

Homework Statement



Given nonzero vectors u, v, and w, use dot product and cross product notation to describe the following.
  1. A vector orthogonal to u X v and u X w
  2. A vector orthogonal to u + v and u - v
  3. A vector of length |u| in the direction of v
  4. The area of the parallelogram determined by u and w

Homework Equations





The Attempt at a Solution



  1. (u X v) X (u X w)
  2. (u + v) X (u - v)
  3. |u|v
  4. |u X w|
 
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  • #2
Check 3. What's the length of |u|v?
 
  • #3
Dick said:
Check 3. What's the length of |u|v?

[STRIKE]If |u| = k, where k is some constant, then the length of |u|v would be kv1 + kv2.[/STRIKE]

Hmm..

I think I see my mistake. It should be [tex]\frac{\vec{u}}{|\vec{u}|}\vec{v}[/tex].
 
Last edited:
  • #4
Dembadon said:
If |u| = k, where k is some constant, then the length of |u|v would be kv1 + kv2.

Hmm..

I think I see my mistake. It should be [tex]\frac{\vec{u}}{|\vec{u}|}\vec{v}[/tex].

you might want to check again
 
  • #5
[itex]\frac{\vec{v}}{|\vec{v}|}[/itex] is a unit vector in the direction of [itex]\vec{v}[/itex]. I need to multiply the unit vector by [itex]|\vec{u}|[/itex].

So, [itex]|\vec{u}|\frac{\vec{v}}{|\vec{v}|}[/itex].
 
  • #6
looks good
 

1. What is the cross product in multi-variable calculus?

The cross product is a mathematical operation that takes two 3-dimensional vectors as input and outputs a vector that is perpendicular to both of the input vectors. In multi-variable calculus, cross products are used to calculate the area of a parallelogram or the volume of a parallelepiped.

2. How is the cross product expressed in multi-variable calculus?

In multi-variable calculus, the cross product is usually expressed using the determinant of a 3x3 matrix. The resulting vector has components that are calculated using the coefficients of the matrix.

3. What is the geometric interpretation of the cross product?

The cross product has a geometric interpretation as the area of a parallelogram formed by the two input vectors. The direction of the resulting vector is perpendicular to this parallelogram, and the magnitude is equal to the area of the parallelogram.

4. What are some applications of cross product expressions in multi-variable calculus?

Cross product expressions are used in many fields, including engineering, physics, and computer graphics. They can be used to calculate torque, determine the orientation of a 3D object, and find the direction of a magnetic field in a given space.

5. How do you calculate the cross product of two vectors in multi-variable calculus?

To calculate the cross product of two vectors in multi-variable calculus, you can use the determinant formula or the component formula. The determinant formula involves taking the determinant of a 3x3 matrix, while the component formula uses the coefficients of the matrix to calculate the components of the resulting vector.

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