Question

the magnitude and direction of two vectors are shown in the diagram. what is the magnitude of their sum? diagram of two vectors with magnitudes 2 and 4, angles 135° and 45° from axes options: 6, 2√5, 20, 8

Explanation:

Step1: Find the components of each vector

For the vector of magnitude \(4\) at \(45^{\circ}\):

• \(x\) - component: \(4\cos45^{\circ}=4\times\frac{\sqrt{2}}{2} = 2\sqrt{2}\)
• \(y\) - component: \(4\sin45^{\circ}=4\times\frac{\sqrt{2}}{2}=2\sqrt{2}\)

For the vector of magnitude \(2\) at \(135^{\circ}\):

• \(x\) - component: \(2\cos135^{\circ}=2\times(-\frac{\sqrt{2}}{2})=-\sqrt{2}\)
• \(y\) - component: \(2\sin135^{\circ}=2\times\frac{\sqrt{2}}{2}=\sqrt{2}\)

Step2: Sum the components

• \(x\) - sum: \(2\sqrt{2}+(-\sqrt{2})=\sqrt{2}\)
• \(y\) - sum: \(2\sqrt{2}+\sqrt{2}=3\sqrt{2}\)

Step3: Find the magnitude of the resultant vector

Using the formula \(|\vec{R}|=\sqrt{x^{2}+y^{2}}\), where \(x = \sqrt{2}\) and \(y=3\sqrt{2}\)

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Answer:

\(2\sqrt{5}\)