When to us dot versus cross product

In summary, dot products are used to measure lengths and angles, while cross products are used to measure areas. The dot product is used in squaring the sum of two vectors because it gives the length of the projection of one vector onto the other, which is needed to compute the area of the parallelogram they span. This is why the dot product is used instead of the cross product in this case. Additionally, the cross product of any vector with itself is 0, making it unnecessary for this calculation.
  • #1
Schmigan
1
0
Hi folks,
When you're squaring the sum of two vectors (v_1 + v_2)^2, why is it that it comes out as v_1 dot v_1 plus 2*v_1 dot v_2 plus v_2 dot v_2? Why do we use the dot product here instead of cross product? I understand that dot product is the multiplication of their parallel components, but it seems arbitrary to use dot rather than cross product in this case
 
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  • #2
dot products: v.w = |v| |w| cos(t), hence these are used to measure lengths and angles.

well i guess it isn't so clear from this description, but cross products are used to measure areas, i.e. |vxw| = |v||w| sin(t).

the reason is that one gives the length of the projection of v onto w, and the other gives the length of the projection of v perpendicular to w, hence gives the height needed to compute area of the parallelogram they span.
 
  • #3
The cross product of any vector with itself is 0. Not much point in doing that!
 

1. When should I use a dot product versus a cross product?

The dot product is used to calculate the scalar projection of one vector onto another, while the cross product is used to calculate the vector perpendicular to two given vectors. Therefore, the dot product is used when you want to find the magnitude of one vector in the direction of another vector, while the cross product is used when you want to find a vector that is perpendicular to both given vectors.

2. How do I know which product to use in a specific situation?

The decision to use the dot product or the cross product depends on what information you are trying to obtain. If you are looking for a scalar quantity, such as work or energy, then the dot product is appropriate. If you are looking for a vector quantity, such as torque or angular momentum, then the cross product is appropriate.

3. Can I use the dot product and cross product interchangeably?

No, the dot product and cross product are two distinct mathematical operations and cannot be used interchangeably. They have different properties and are used for different purposes, as mentioned in the answers to the previous questions.

4. What are the mathematical formulas for the dot product and cross product?

The dot product of two vectors a and b is given by a · b = |a||b| cos θ, where θ is the angle between the two vectors. The cross product of two vectors a and b is given by a x b = |a||b| sin θ n, where θ is the angle between the two vectors and n is the unit vector perpendicular to both a and b.

5. Are there any real-life applications of the dot product and cross product?

Yes, the dot product and cross product have various real-life applications in physics, engineering, and computer graphics. Some examples include calculating the work done by a force on an object, finding the torque on a rotating object, and determining the direction of a 3D rotation in computer graphics.

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