Question

In Exercises \(65-68,\) find the magnitude of \(v\). \(\mathbf{v}=\mathbf{i}-2 \mathbf{j}-3 \mathbf{k}\)

Short answer

The magnitude of the vector \(v\) is \(\sqrt{14}\).

Step-by-step solution

Identify the components of the vector

The components of the vector \(v\) are: \(i = 1, j = -2, k = -3\)

Substitute the components into the magnitude formula

The formula for the magnitude of a vector in three-dimensional space is \(\sqrt{(v_i)^2 + (v_j)^2 + (v_k)^2}\). Substituting \(i = 1, j = -2, k = -3\) into the formula results in \(\|v\| = \sqrt{(1)^2 + (-2)^2 + (-3)^2}\).

Calculate the magnitude

Perform the calculation to find the magnitude of vector \(v\): \(\|v\| = \sqrt{1 + 4 + 9} = \sqrt{14}\).