A particle moves $21\, m$ along the vector $6\hat i + 2\hat j + 3\hat k$ , then $14\, m$ along the vector $3\hat i - 2\hat j + 6\hat k$ . Its total displacement (in meters) is

and direction of the vectors $\hat{ i }+\hat{ j }$, and $\hat{ i }-\hat{ j }$ ? What are the components of a vector $A =2 \hat{ i }+3 \hat{ j }$ along the directions of $\hat{ i }+\hat{ j }$ and $\hat{ i }-\hat{ j } ?$

Three particles, located initially on the vertices of an equilateral triangle of side $L,$ start moving with a constant tangential acceleration towards each other in a cyclic manner, forming spiral loci that coverage at the centroid of the triangle. The length of one such spiral locus will be

A man on a rectilinearly moving cart, facing the direction of motion, throws a ball straight up with respect to himself

The coordinates of a moving particle at any time $‘t’$ are given by $ x = \alpha t^3$ and $y = \beta t^3$. The speed of the particle at time $‘t’$ is given by