Umakant Kondapure, Collin Fernandes, Nipun Bhatia, Vikram Bathula and, Ketki Deshpande Solutions for Chapter: Scalars and Vectors, Exercise 2: Critical Thinking

If $\overrightarrow{A}·\overrightarrow{B}=0$ and $\overrightarrow{A}\times \overrightarrow{B}=0$, then which of the following conditions is necessary?

If the ratio of the dot product of two vectors and cross product of same two vectors is $\sqrt{3}$, the two vectors make an angle of,

Select the wrong one.

If $\overrightarrow{A}$ and $\overrightarrow{B}$ are two vectors then, $\left(\overrightarrow{A}+\overrightarrow{B}\right)\times \left(\overrightarrow{A}-\overrightarrow{B}\right)$ is,

Given $\overrightarrow{P}·\left(\overrightarrow{P}+\overrightarrow{Q}\right)=P^2$, then the angle between $\overrightarrow{P}$ and $\overrightarrow{Q}$ is,

Assertion: If the dot product and the cross product of $\overrightarrow{A}$ and $\overrightarrow{B}$ are zero, it implies that one of the vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ must be a null vector.

Reason: Null vector is a vector with zero magnitude.

Assertion: $\overrightarrow{A}\times \overrightarrow{B}$ is perpendicular to both $\overrightarrow{A}+\overrightarrow{B}$ as well as $\overrightarrow{A}-\overrightarrow{B}$.

Reason: $\overrightarrow{A}+\overrightarrow{B}$ as well as $\overrightarrow{A}-\overrightarrow{B}$ lie in the plane containing $\overrightarrow{A}$ and $\overrightarrow{B}$ but $\overrightarrow{A}\times \overrightarrow{B}$ lies perpendicular to the plane containing $\overrightarrow{A}$ and $\overrightarrow{B}$.

A small ball is hung by a string fixed to a wall. The ball is pushed away from the wall. The forces acting on the ball are shown in the diagram. Which of the following statements is wrong?