Question

Find the magnitude of \(v\). Initial point of \(\mathbf{v}:(1,-3,4)\) Terminal point of \(\mathbf{v}:(1,0,-1)\)

Short answer

The magnitude of vector \(v\), given the initial point A(1,-3,4) and terminal point B(1,0,-1), is calculated as \(|\mathbf{v}| = \sqrt{(0)^2 + (3)^2 + (-5)^2} = \sqrt{34}\).

Step-by-step solution

Determine Vector From Initial and Terminal Points

Begin by determining the vector \(\mathbf{v}\) between the given initial point A(1,-3,4) and terminal point B(1,0,-1). A vector is defined as a directed line segment from an initial point (or origin) to a terminal point (or end). Apply the formula: \(\mathbf{v} = B - A\) in separate x, y, and z components. Or, \(\mathbf{v}= (x_2-x_1, y_2-y_1, z_2-z_1)\). Here, \(x_1, y_1, z_1\) are the coordinates of the initial point and \(x_2, y_2, z_2\) are the coordinates of the terminal point.

Calculate Vector v

Now, using the above formula, calculate the vector \(\mathbf{v}\): \(\mathbf{v} =(1-1, 0--3, -1-4) = (0,3,-5)\). So, the vector \(\mathbf{v}\) is (0,3,-5).

Compute Magnitude of Vector v

Once the vector \(\mathbf{v}\) is computed, the magnitude (or length) of the vector is calculated, using the formula: \[|\mathbf{v}| = \sqrt{(v_x)^2 + (v_y)^2 + (v_z)^2}\], where \(v_x, v_y, v_z\) are the x, y, z values of vector \(\mathbf{v}\). Substituting into the formula renders the magnitude of \(\mathbf{v}\).