A vector $\vec r=a \hat i+b \hat j$ is equally inclined to both x and y axes. If the magnitude of the vector is 2 units, then what are the values of a and b respectively?

Consider the following statements in respect of a vector $\vec c=\vec a+\vec b$, where $|\vec a|=|\vec b|\ne0$:

1. $\vec c$ is perpendicular to $(\vec a-\vec b).$

2. $\vec c$ is perpendicular to $\vec a \times \vec b.$

Which of the above statement is/are correct?

If $\vec a \:and\: \vec b$ are two vectors such that $|\vec a + \vec b|= |\vec a - \vec b|=4,$ then which one of the following is correct?

If $\vec a, \vec b\:and \: \vec c$ are coplanar, then what is $(2\vec a\times 3\vec b)\cdot4\vec c+(5\vec b\times 3\vec c)\cdot6\vec a$ equal to?