Physics - Vectors Question with Solution | TestHub

Statement-1: The scalar product (dot product) of two non-zero vectors $\vec{A}$ and $\vec{B}$ is given by $\vec{A}\cdot \vec{B}=|\vec{A}||\vec{B}|\cos \theta$. If $\vec{A}\cdot \vec{B}=0$, then $|\vec{A}||\vec{B}|\cos \theta =0$. Since $\vec{A}$ and $\vec{B}$ are non-zero, $|\vec{A}|\ne 0$ and $|\vec{B}|\ne 0$.

Therefore, $\cos \theta =0$, which implies $\theta =90^{\circ }$ (or $\pi /2$ radians). This means the vectors are perpendicular. Conversely, if they are perpendicular, $\theta =90^{\circ }$, so $\cos \theta =0$, and $\vec{A}\cdot \vec{B}=0$. Thus, Statement-1 is true.

Statement-2: The magnitude of the vector product (cross product) of two non-zero vectors $\vec{A}$ and $\vec{B}$ is given by $|\vec{A}\times \vec{B}|=|\vec{A}||\vec{B}|\sin \theta$. For this magnitude to be maximum, $\sin \theta$ must be maximum. The maximum value of $\sin \theta$ is $1$, which occurs when $\theta =90^{\circ }$. When vectors are parallel, $\theta =0^{\circ }$ or $\theta =180^{\circ }$, for which $\sin \theta =0$, making the magnitude of the vector product zero (minimum). Thus, Statement-2 is false.

Statement-1 is true and Statement-2 is false.