Properties of Cross Product Physics Notes
Properties of cross product:
1. Vector products do not show commutative property:
2. Vector Product of Two Parallel Vectors:
Let \(\vec{A}\) and \(\vec{B}\) be two parallel vector. So, the angle between the two vectors, θ = 0°.
From vector products,
So, the vector product of two parallel vectors is a zero product.
3. Vector Product of Equal Vectors: Equal vectors are also parallel. So, the angle between them will also be zero. So, θ = 0°.
∴ \(\vec{A} \times \vec{A}\) = AA sin 0° = 0
So, the vector product of equal vectors is a zero vector.
∴ î × î = ĵ × ĵ = k̂ × k̂ = 0 …(iv)
4. Vector Product of Perpendicular:
Vectors: Let us assume that \(\vec{A}\) and \(\vec{B}\) are two vectors perpendicular to each other. So, the angle between them will be 90°, i.e., θ = 90°.
Then,
Here, the direction of ñ is according to the right-hand rule, in the direction perpendicular to the plane of \(\vec{A}\) and \(\vec{B}\). So, in this way,
If î, ĵ, and k̂ are in the cyclic order (clockwise) then the vector product is positive and if not in cyclic then the product will be negative.
5. Vector Product in the form of a Determinant: If the vectors are expressed in terms of unit vectors î, ĵ and k̂ in the X, Y, and Z directions, then the vectors can be expressed as:
Using equations (iv) and (v), we have
It can be re-written in the form of determinants
The magnitude of cross product:
6. Cross product of two vectors is distributive:
i. e., \(\vec{A} \times(\vec{B}+\vec{C})=\vec{A} \times \vec{B}+\vec{B} \times \vec{C}\)
7. Cross product of two vectors is associative i.e.,
\((\vec{A}+\vec{B}) \times(\vec{C}+\vec{D})=\vec{A} \times \vec{C}+\vec{A} \times \vec{D}+\vec{B} \times \vec{C}+\vec{B} \times \vec{D}\)
8. Examples of Some Physical Quantities Obtained from Vector Product
(a) Let \(\vec{L}\) and \(\vec{B}\) respectively be the adjacent sides of a parallelogram, then the area of it will be:
\(\vec{A}=\vec{L} \times \vec{B}\)
(b) Torque (\(\vec{τ}\)), displacement (\(\vec{r}\)) and force (\(\vec{F}\)) are associated as:
\(\vec{\tau}=\vec{r} \times \vec{F}\)
i.e., torque (\(\vec{τ}\)) is the cross product of displacement and force.
(c) Angular momentum (J), displacement (\(\vec{r}\)) and linear momentum (\(\vec{p}\)) are associated as:
\(\vec{J}=\vec{r} \times \vec{p}\)
i. e., angular momentum is equal to the cross-product of displacement and linear momentum.
(d) Angular velocity (ω), linear velocity (v) and displacement (r) are associated as :
\(\vec{v}=\vec{\omega} \times \vec{r}\)
i.e., linear velocity is equal\o the cross-product of angular velocity and displacemerí.
(e) The vectors of adjacent sidef a triangle are and b the area of triangle:
\(\vec{A}=\frac{1}{2}|\vec{a} \times \vec{b}|\)