the magnitude and direction of two vectors are shown in the diagram. what is the magnitude of their sum…

Explanation:

Step1: Resolve vectors into x and y components

For the vector with magnitude 2 and angle (135^\circ) (from positive x - axis): The x - component (v_{1x}=2\cos135^\circ), and (\cos135^\circ =-\frac{\sqrt{2}}{2}), so (v_{1x}=2\times(-\frac{\sqrt{2}}{2})=-\sqrt{2}). The y - component (v_{1y}=2\sin135^\circ), and (\sin135^\circ=\frac{\sqrt{2}}{2}), so (v_{1y}=2\times\frac{\sqrt{2}}{2}=\sqrt{2}).

For the vector with magnitude 4 and angle (45^\circ) (from positive x - axis): The x - component (v_{2x}=4\cos45^\circ), and (\cos45^\circ=\frac{\sqrt{2}}{2}), so (v_{2x}=4\times\frac{\sqrt{2}}{2}=2\sqrt{2}). The y - component (v_{2y}=4\sin45^\circ), and (\sin45^\circ=\frac{\sqrt{2}}{2}), so (v_{2y}=4\times\frac{\sqrt{2}}{2}=2\sqrt{2}).

Step2: Find the resultant x and y components

The resultant x - component (V_x = v_{1x}+v_{2x}=-\sqrt{2}+2\sqrt{2}=\sqrt{2}). The resultant y - component (V_y = v_{1y}+v_{2y}=\sqrt{2}+2\sqrt{2}=3\sqrt{2}).

Step3: Calculate the magnitude of the resultant vector

The magnitude of a vector (\vec{V}) with components (V_x) and (V_y) is given by (|\vec{V}|=\sqrt{V_x^{2}+V_y^{2}}). Substitute (V_x = \sqrt{2}) and (V_y = 3\sqrt{2}) into the formula: [ \begin{align*} |\vec{V}|&=\sqrt{(\sqrt{2})^{2}+(3\sqrt{2})^{2}}\ &=\sqrt{2 + 18}\ &=\sqrt{20}\ &=2\sqrt{5} \end{align*} ]

Answer: (2\sqrt{5})