the magnitude and direction of two vectors are shown in the diagram. what is the magnitude of their sum? 6 8…

Explanation:

Step1: Resolve vectors into components

Let the first vector $\vec{A}$ with magnitude $A = 4$ at an angle $\theta_1=45^{\circ}$ and the second vector $\vec{B}$ with magnitude $B = 2$ at an angle $\theta_2 = 135^{\circ}$. The $x$-components are: $A_x=A\cos\theta_1=4\cos45^{\circ}=4\times\frac{\sqrt{2}}{2}=2\sqrt{2}$ $B_x=B\cos\theta_2=2\cos135^{\circ}=2\times(-\frac{\sqrt{2}}{2})=-\sqrt{2}$ The $y$-components are: $A_y=A\sin\theta_1=4\sin45^{\circ}=4\times\frac{\sqrt{2}}{2}=2\sqrt{2}$ $B_y=B\sin\theta_2=2\sin135^{\circ}=2\times\frac{\sqrt{2}}{2}=\sqrt{2}$

Step2: Calculate the sum of $x$-components and $y$-components

$R_x=A_x + B_x=2\sqrt{2}-\sqrt{2}=\sqrt{2}$ $R_y=A_y + B_y=2\sqrt{2}+\sqrt{2}=3\sqrt{2}$

Step3: Calculate the magnitude of the resultant vector

The magnitude of the resultant vector $\vec{R}$ is given by $R=\sqrt{R_x^{2}+R_y^{2}}$. $R=\sqrt{(\sqrt{2})^{2}+(3\sqrt{2})^{2}}=\sqrt{2 + 18}=\sqrt{20}=2\sqrt{5}$

Answer:

$2\sqrt{5}$