Vectors Questions and Answers

The position vectors of the points $A$, $B$ and $C$ relative to the origin $O$ are $a = 3i - j$, $b = 2i + j + 3k$ and $c = i - j + 6k$. Find the value of $a \cdot (b \times c)$.

For three non-zero vectors $a$, $b$ and $c$, the scalar triple product $a \cdot (b \times c)$ measures which quantity?

A line has direction $d = 3i - k$ and passes through the point $(8, 1, -1)$. A plane has equation $r \cdot n = 5$, where $n = -i + 10j - 3k$. Which pair of checks shows that the line lies in the plane?

Find the angle between the planes $r \cdot (-i + 10j - 3k) = 5$ and $r \cdot (i + j + 3k) = 6$.