91Ó°ÊÓ

We are given the following vectors: \(\mathbf{u}=\langle 3,-4\rangle\), \(\mathbf{v}=\langle 1,1\rangle,\) and \(\mathbf{w}=\langle 1,0\rangle\). We need to find the vector \(2\mathbf{u} + 3\mathbf{v} - 4\mathbf{w}\). This can be done by adding the corresponding components of the vectors and then multiplying them by the given constants: $$ 2\mathbf{u} + 3\mathbf{v} - 4\mathbf{w} = 2\langle 3,-4\rangle + 3\langle 1,1\rangle - 4\langle 1,0\rangle $$ Now, let's calculate the resulting vector: $$ 2\langle 3,-4\rangle + 3\langle 1,1\rangle - 4\langle 1,0\rangle = \langle 2(3), 2(-4)\rangle + \langle 3(1), 3(1)\rangle - \langle 4(1), 4(0)\rangle $$ $$ = \langle 6,-8 \rangle + \langle 3,3 \rangle - \langle 4,0 \rangle $$ $$ = \langle 6+3-4,-8+3+0 \rangle = \langle 5,-5 \rangle $$ So, the resulting vector is \(\langle 5, -5 \rangle\).

Now that we have the resulting vector \(\langle 5,-5\rangle\), we can calculate its magnitude. The magnitude of a 2-dimensional vector \(\langle x,y \rangle\) is given by the formula: $$ |\mathbf{v}| = \sqrt{x^2 + y^2} $$ Thus, the magnitude of our vector \(\langle 5,-5\rangle\) is: $$ |\langle 5,-5\rangle| = \sqrt{ (5)^2 + (-5)^2 } $$ $$ = \sqrt{ 25 + 25 } = \sqrt{50} $$ So, the magnitude of the resulting vector is \(\sqrt{50}\).