Question 1

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three mutually perpendicular vectors each of unit magnitude. If $\vec{A}=\vec{a}+\vec{b}+\vec{c}$, $\vec{B}=\vec{a}-\vec{b}+\vec{c}$ and $\vec{C}=\vec{a}-\vec{b}-\vec{c}$, then which one of the following is correct?

Question 2

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be unit vectors such that $\vec{a}\times\vec{b}$ is perpendicular to $\vec{c}$. If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then which of the following is/are correct? 1. $\vec{a}\times\vec{b} = \sin\theta\,\vec{c}$ 2. $\vec{a}\cdot(\vec{b}\times\vec{c}) = 0$

Question 3

The dot product $(\hat i+2\hat j+3\hat k)\cdot(2\hat i+3\hat j+4\hat k)$ is

Question 4

Let $\theta$ be the angle between two unit vectors $\vec{a}$ and $\vec{b}$. If $\vec{a} + 2\vec{b}$ is perpendicular to $5\vec{a} - 4\vec{b}$, then what is $\cos\theta + \cos 2\theta$ equal to?

Question 5

Let $ABCDEF$ be a regular hexagon. If $\vec{AD} = m\vec{BC}$ and $\vec{CF} = n\vec{AB}$, then what is $mn$ equal to?