Question

the magnitude and direction of two vectors are shown in the diagram. what is the magnitude of their sum? 6 8 $2\sqrt{5}$ 20

Explanation:

Step1: Find x,y components of vector 4

$V_{1x} = 4\cos(45^\circ) = 4\times\frac{\sqrt{2}}{2}=2\sqrt{2}$
$V_{1y} = 4\sin(45^\circ) = 4\times\frac{\sqrt{2}}{2}=2\sqrt{2}$

Step2: Find x,y components of vector 2

$V_{2x} = 2\cos(135^\circ) = 2\times(-\frac{\sqrt{2}}{2})=-\sqrt{2}$
$V_{2y} = 2\sin(135^\circ) = 2\times\frac{\sqrt{2}}{2}=\sqrt{2}$

Step3: Sum x,y components

$V_{totalx}=2\sqrt{2}+(-\sqrt{2})=\sqrt{2}$
$V_{totaly}=2\sqrt{2}+\sqrt{2}=3\sqrt{2}$

Step4: Calculate magnitude of resultant

$|V_{total}|=\sqrt{(\sqrt{2})^2+(3\sqrt{2})^2}=\sqrt{2 + 18}=\sqrt{20}=2\sqrt{5}$

Answer:

$2\sqrt{5}$