Answer
$ \|\mathbf{a}_N\| = 6\sqrt{2}$
Work Step by Step
Step 1 Given: \[ \|\mathbf{a}(1)\| = 9 \] \[ \|\mathbf{a}(1)\| = 9; \quad \mathbf{a}_T(1)\mathbf{T}(1) = 2\mathbf{i} - 2\mathbf{j} + \mathbf{k} \] Step 2 Since: \[ \mathbf{a} = \mathbf{a}_T\mathbf{T} + \mathbf{a}_N\mathbf{N} \] Then: \[ \|\mathbf{a}\|^2 = \|\mathbf{a}_T\|^2 + \|\mathbf{a}_N\|^2 = 9 + \|\mathbf{a}_N\|^2 \] \[ 81 = 9 + \|\mathbf{a}_N\|^2 \] Hence: \[ \|\mathbf{a}_N\| = \sqrt{72}. \] Result \[ \|\mathbf{a}_N\| = 6\sqrt{2} \]
