If the vector $2\hat{i} + 3\hat{j} - \hat{k}$ is perpendicular to the vector $-4\hat{i} - 6\hat{j} + \lambda\hat{k}$,find the value of $\lambda$.

Explain the geometrical interpretation of the scalar product of two vectors.

Show that the area of the triangle formed by the vectors $\vec{a}$ and $\vec{b}$ is one-half of the magnitude of $\vec{a} \times \vec{b}$.

$P$ and $Q$ are two non-zero vectors inclined to each other at an angle $\theta$. The component of $Q$ in the direction of $P$ is:

Vectors $a \hat{i} + b \hat{j} + \hat{k}$ and $2 \hat{i} - 3 \hat{j} + 4 \hat{k}$ are perpendicular to each other. Given that $3a + 2b = 7$,and the ratio of $a$ to $b$ is $\frac{x}{2}$,find the value of $x$.

What is the unit vector perpendicular to the vectors $2\hat i + 2\hat j - \hat k$ and $6\hat i - 3\hat j + 2\hat k$?