Confusion about the angle between two vectors in a cross product

[tbn032]

The magnitude of cross product is defined of vector A⃗ and B⃗ as |A⃗×B⃗|=|A⃗||B⃗|sinθ where θ is defined as the angle between the two vector and 0≤θ≤π.the domain of θ is defined 0≤θ≤π so that the value of sinθ remains positive and thus the value of the magnitude |A⃗||B⃗|sinθ also remain positive (magnitude cannot be negative).

But if A⃗=1i^ and B⃗ =-j^ then the angle between these vector would be -π/2 but the magnitude of |A⃗×B⃗|=|A⃗||B⃗|sinθ where the domain of θ is defined 0≤θ≤π.

how can this angle(-π/2) be incorporated in the formula so that the magnitude of cross product of these vectors could be found. In general, how can the θ whose value is π≤θ≤2π incorporated in the formula so that the magnitude of the cross product of the vectors can be calculated.

The angle between vector A⃗ and B⃗ can be θ=π/2 if we measure anti-clockwise (From vector B⃗ to A⃗ and θ=-π/2 if we measure the angle clockwise(from vector A⃗ to B⃗).how can I say which angle to pick.

 

[PeroK]

The magnitude of cross product is defined of vector A⃗ and B⃗ as |A⃗×B⃗|=|A⃗||B⃗|sinθ where θ is defined as the angle between the two vector and 0≤θ≤π.the domain of θ is defined 0≤θ≤π so that the value of sinθ remains positive and thus the value of the magnitude |A⃗||B⃗|sinθ also remain positive (magnitude cannot be negative).

But if A⃗=1i^ and B⃗ =-j^ then the angle between these vector would be -π/2 but the magnitude of |A⃗×B⃗|=|A⃗||B⃗|sinθ where the domain of θ is defined 0≤θ≤π.

how can this angle(-π/2) be incorporated in the formula so that the magnitude of cross product of these vectors could be found. In general, how can the θ whose value is π≤θ≤2π incorporated in the formula so that the magnitude of the cross product of the vectors can be calculated.

The angle between vector A⃗ and B⃗ can be θ=π/2 if we measure anti-clockwise (From vector B⃗ to A⃗ and θ=-π/2 if we measure the angle clockwise(from vector A⃗ to B⃗).how can I say which angle to pick.

We have $\vec A \times \vec B = |A||B|\sin \theta$, hence $|\vec A \times \vec B| = |A||B||\sin \theta|$. Or, when dealing with the modulus you could change your definition of $\theta$ to lie in the range $[0, \pi)$.

 

[Mark44]

The formula shown here (https://en.wikipedia.org/wiki/Cross_product, in the Properties section) gives the magnitude of the cross product as $$||\vec a \times \vec b|| = ||\vec a|| ||\vec b|| |\sin(\theta)|$$

BTW, in what you wrote there are some extra symbols whose purpose I don't understand.

But if A⃗=1i^ and B⃗ =-j^

What is the meaning of the square after A?
What is the meaning of the caret after j?

 

[tbn032]

BTW, in what you wrote there are some extra symbols whose purpose I don't understand.


What is the meaning of the square after A?
What is the meaning of the caret after j?

I just meant i cap and j cap (unit vectors along x-axis and y-axis, respectively)

 

[tbn032]

Or, when dealing with the modulus you could change your definition of $\theta$ to lie in the range $[0, \pi)$.

I think the theta will lie in the range $[0, \pi]$ instead of $[0, \pi)$.

 

[berkeman]

I just meant i cap and j cap (unit vectors along x-axis and y-axis, respectively)

Please have a look at the "LaTeX Guide" link below the Edit window. That will help you to post your math equations in a much more readable form. Thank you.

 

[robphy]

Note that the cross-product is often associated with the oriented parallelogram formed by its factors. (It might help to think "bi-vector. https://en.wikipedia.org/wiki/Bivector )
The magnitude of the cross-product is equal to the area of that parallelogram.
With the tails of the vectors together, the interior angle is a signed-angle whose magnitude is not larger than $\pi$.

 

[tbn032]

The magnitude of the cross-product is equal to the area of that parallelogram.
With the tails of the vectors together, the interior angle is a signed-angle whose magnitude is not larger than $\pi$.

In many of the definition of cross product, I have seen the $\theta$(angle between the two vector) is in the range of 0≤$\theta$≤π.

The angle between the two vector can be measured in two ways, clockwise and anti-clockwise. The clockwise wise measurement is generally taken to be negative and the anti-clockwise wise measurement is generally taken to be positive.

How can the $\theta$ always lie in the 0≤$\theta$≤π.for example, take two vectors $\vec A$ =1$\hat i$ and $\vec B$ =-1$\hat j$.the angle between these vectors could be measured -π/2 and π/2 if we measure it clockwise and anticlockwise respectively. How can the -π/2 incorporated in the range 0≤$\theta$≤π.

is it the case that we ignore clockwise measurement when measuring the angle between vector or is it the case that the $\theta$ which is present in the formula |A⃗×B⃗|=|A⃗||B⃗|sin$\theta$ is just the magnitude of the angle present between the two vector and direction(clockwise or anti-clockwise) is not considered($\theta$=|angle between the vectors|)