Three Dimensional Geometry

What is the distance between the points A(1, 2, 3) and B(4, 6, 8)?

If the position vectors of points A and B are \(\vec{a} = 2\hat{i} + 3\hat{j} + \hat{k}\) and \(\vec{b} = -\hat{i} + 4\hat{j} + 2\hat{k}\), what is the vector \(\vec{AB}\)?

Find the equation of the plane passing through point P(1, 2, 3) and perpendicular to the vector \(\vec{n} = 2\hat{i} - 3\hat{j} + 4\hat{k}\).

Find the angle between the vectors \(\vec{a} = 3\hat{i} - \hat{j} + 2\hat{k}\) and \(\vec{b} = -\hat{i} + 4\hat{j} + 2\hat{k}\).

Find the foot of the perpendicular drawn from point P(2, -1, 3) to the plane \(3x - y + 4z = 12\).

Find the equation of the line passing through points A(1, 2, 3) and B(4, 0, -1).

If \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\) and \(\vec{b} = 4\hat{i} - \hat{j} + 2\hat{k}\), find the vector perpendicular to both \(\vec{a}\) and \(\vec{b}\).

Find the shortest distance between the skew lines: \(\frac{x-1}{2} = \frac{y+1}{-1} = \frac{z}{3}\) and \(\frac{x}{1} = \frac{y-2}{4} = \frac{z-1}{-2}\).

Find the volume of the parallelepiped formed by vectors \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\), \(\vec{b} = 4\hat{i} - \hat{j} + 2\hat{k}\), and \(\vec{c} = 2\hat{i} + \hat{j} - \hat{k}\).