Let P⃗ be the linear momentum of a particle whose position vector is r⃗ with respect to the orig

step 1 In this question we have given linear momentum of a particle and we have given the angular momen

Let P⃗ be the linear momentum of a particle whose position...

Let $\vec{P}$ be the linear momentum of a particle whose position vector is $\vec{r}$ with respect to the origin and $\vec{L}$ be the angular momentum of this particle about the origin, then (A) $\vec{r} \cdot \vec{L}=0$ and $\vec{P} \cdot \vec{L}=0$ (B) $\vec{r} \cdot \vec{L}=0$ and $\vec{P} \cdot \vec{L} \neq 0$ (C) $\vec{r} \cdot \vec{L} \neq 0$ and $\vec{P} \cdot \vec{L}=0$ (D) $\vec{r} \cdot \vec{L} \neq 0$ and $\vec{P} \cdot \vec{L} \neq 0$

Let $\vec{P}$ be the linear momentum of a particle whose position vector is $\vec{r}$ with respect to the origin and $\vec{L}$ be the angular momentum of this particle about the origin, then
(A) $\vec{r} \cdot \vec{L}=0$ and $\vec{P} \cdot \vec{L}=0$
(B) $\vec{r} \cdot \vec{L}=0$ and $\vec{P} \cdot \vec{L} \neq 0$
(C) $\vec{r} \cdot \vec{L} \neq 0$ and $\vec{P} \cdot \vec{L}=0$
(D) $\vec{r} \cdot \vec{L} \neq 0$ and $\vec{P} \cdot \vec{L} \neq 0$

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Key Concepts

Linear Momentum

Linear momentum is a vector quantity defined as the product of a particle's mass and its velocity. It describes the quantity of motion an object has and is essential in both the conservation of momentum and dynamics analysis in physics.

Angular Momentum

Angular momentum is a vector quantity representing the rotational inertia and the rotational motion of a particle about a point. It is defined as the cross product of the position vector and the linear momentum vector, which inherently introduces properties relating to direction and magnitude in rotational dynamics.

Cross Product Properties

The cross product of two vectors results in a new vector that is perpendicular to the plane containing the original vectors. This property is key in the definition of angular momentum, as it guarantees that the angular momentum vector is orthogonal to both the position vector and the momentum vector.

Orthogonality in Vector Products

Because the cross product yields a vector perpendicular to the original vectors, any dot product of an original vector with the cross product result will be zero. This orthogonality is why the dot products of the position vector with angular momentum, and of linear momentum with angular momentum, vanish in the context discussed.

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