Video: Finding the Magnitude of a Vector

Given that |𝐀| = |𝐁|, 𝐀 β‹… 𝐁 = 69 , and the measure of the angle between 𝐀 and 𝐁 is 60Β°, determine |𝐀| to the nearest hundredth.

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Video Transcript

Given that the magnitude of 𝐀 is equal to the magnitude of 𝐁, 𝐀 dot 𝐁 is equal to 69, and the measure of the angle between 𝐀 and 𝐁 is 60 degrees, determine the magnitude of 𝐀 to the nearest hundredth.

We’re dealing with two vectors in this question, vector 𝐀 and vector 𝐁. We’re told the angle between these vectors is equal to 60 degrees. When we’re thinking about angles between two vectors, we should be thinking about a scalar product. This says that if 𝐀 and 𝐁 are two vectors, cos of πœƒ is equal to π‘Ž dot 𝑏 over the magnitude of π‘Ž times the magnitude of 𝑏.

Now in this case, πœƒ is equal to 60. So we have cos of 60 degrees. We’re told that the dot product of 𝐀 and 𝐁 is equal to 69. Then this is all over the magnitude of 𝐀 times the magnitude of 𝐁. Now of course, we know that cos of 60 degrees is equal to one-half. But we’re also told that the magnitude of 𝐀 is equal to the magnitude of 𝐁. So we can rewrite the magnitude of 𝐀 times the magnitude of 𝐁 as the magnitude of 𝐀 squared.

To solve for this equation for the magnitude of 𝐀, we’ll begin by multiplying both sides by the magnitude of 𝐀 squared. And we have one-half times the magnitude of 𝐀 squared equals 69. Next, we’ll multiply through by two. And we find the magnitude of 𝐀 squared to be equal to 138.

Our final step is going to be to find the square root of both sides. Now normally, we would find both the positive and negative square root of 138. But the magnitude must be a positive value. So we’re only gonna find the positive square root of 138. Well, the positive square root of 138 is 11.7473 and so on. Rounding this to the nearest one hundredth, we find the magnitude of 𝐀 to be equal to 11.75.

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