Question

In Exercises \(41-44,\) find the component form and magnitude of the vector \(u\) with the given initial and terminal points. Then find a unit vector in the direction of \(\mathbf{u}\). \(\frac{\text { Initial Point }}{(3,2,0)}\) \(\frac{\text { Terminal Point }}{(4,1,6)}\)

Short answer

The component form of the vector \(\mathbf{u}\) is (1, -1, 6), the magnitude is \(\sqrt{38}\), and a unit vector in the direction of \(\mathbf{u}\) is (1/\sqrt{38}, -1/\sqrt{38}, 6/\sqrt{38}).

Step-by-step solution

Find the Component Form of the Vector

The component form of the vector \(\mathbf{u}\) is determined by subtracting the coordinates of the initial point from the terminal point: \( (4-3, 1-2, 6-0) = (1, -1, 6) \)

Calculate the Magnitude of the Vector

The magnitude of \(\mathbf{u}\) is calculated using the Pythagorean theorem: \( \sqrt{(1)^2 + (-1)^2 + (6)^2} = \sqrt{38} \)

Find the Unit Vector

The unit vector in the direction of \(\mathbf{u}\) is found by dividing each component of the vector by its magnitude: \( (1/\sqrt{38}, -1/\sqrt{38}, 6/\sqrt{38}) \)