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

Prove the associative law of vector addition.

If $\left| {{{\vec v}_1} + {{\vec v}_2}} \right| = \left| {{{\vec v}_1} - {{\vec v}_2}} \right|$ and ${{{\vec v}_1}}$ and ${{{\vec v}_2}}$ are finite, then

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

The vectors $\overrightarrow A $ and $\overrightarrow B$  lie in a plane. Another vector $\overrightarrow C $ lies outside this plane. The  resultant $\overrightarrow A + \overrightarrow B + \overrightarrow C$ of these three vectors

Two forces $F_1 = 3N$ at $0^o$ and $F_2 = 5N$ at $60^o$ act on a body. Then a single force that would balance the two forces must have a magnitude of .......... $N$