If $|\vec{a}|=4$ and $|\vec{b}|=5$,then the values of $k$ for which $\vec{a}+k \vec{b}$ is perpendicular to $\vec{a}-k \vec{b}$ are

If vectors $\vec{a}$ and $\vec{b}$ have the same magnitude, the angle between them is $60^\circ$, and their scalar product is $1/2$, then $|\vec{a}|$ is:

If $|a \cdot b| = 3$ and $|a \times b| = 4$,then the angle between $a$ and $b$ is

If $p$-th, $q$-th, and $r$-th terms of a geometric progression are the positive numbers $a, b,$ and $c$ respectively, then the angle between the vectors $(\log a^2) i + (\log b^2) j + (\log c^2) k$ and $(q-r) i + (r-p) j + (p-q) k$ is

The orthogonal projection of vector $\vec{a}$ on vector $\vec{b}$ is:

Find a unit vector in the $xy$-plane which makes an angle of $45^{\circ}$ with the vector $i + j$ and an angle of $60^{\circ}$ with the vector $3i - 4j$.