The position vectors of points $A, B, C$ and $D$ are $\vec A = 3\hat i + 4\hat j + 5\hat k,\,\vec B = 4\hat i + 5\hat j + 6\hat k,\,\vec C = 7\hat i + 9\hat j + 3\hat k$ and $\vec D = 4\hat i + 6\hat j$ then the displacement vectors $\overrightarrow {AB} $ and $\overrightarrow {CD} $ are

If the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors, the angle between these vectors is ........ $^o$

Magnitude of vector which comes on addition of two vectors, $6\hat i + 7\hat j$ and $3\hat i + 4\hat j$ is

Two vectors having equal magnitudes of $x\, units$ acting at an angle of $45^o$ have resultant $\sqrt {\left( {2 + \sqrt 2 } \right)} $ $units$. The value of $x$ is

For the resultant of the two vectors to be maximum, what must be the angle between them....... $^o$

A cyclist starts from the centre $O$ of a circular park of radius $1\; km$, reaches the edge $P$ of the park, then cycles along the circumference, and returns to the centre along $QO$ as shown in Figure. If the round trip takes $10 \;min$, what is the

$(a)$ net displacement,

$(b)$ average velocity, and

$(c)$ average speed of the cyclist ?