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Chapter 6: Problem 12

Use the dot product to find the angle between the vectors (3,-5) and (-4,3) .

Short Answer

Expert verified
The angle between the vectors \(a = (3, -5)\) and \(b = (-4, 3)\) is approximately \(127.31掳\).

Step by step solution

01

Calculate the dot product

To calculate the dot product of two vectors, we can multiply their corresponding components and then add the products. Given the vectors \(a = (3, -5)\) and \(b = (-4, 3)\), their dot product can be found using the formula \(a \cdot b = a_x b_x + a_y b_y\), which can be calculated as follows: \[a \cdot b = (3)(-4) + (-5)(3) = -12 - 15 = -27\]
02

Calculate the magnitude of each vector

To find the angle between the two vectors, we also need to know their magnitudes. Recall that the magnitude of a vector can be calculated using the square root of the sum of its component's squares. So, for the given vectors: \[\|a\| = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}\] \[\|b\| = \sqrt{(-4)^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\]
03

Apply the dot product formula to find the angle

Finally, we can use the dot product formula to find the angle (\(\theta\)) between the two vectors: \[a \cdot b = \|a\|\|b\|\cos\theta\] Plugging in the calculated values: \[-27 = \sqrt{34} \cdot 5 \cos\theta\] Now, we can solve for \(\theta\): \[\cos\theta = \frac{-27}{\sqrt{34} \cdot 5}\] \[\theta = \arccos{\frac{-27}{\sqrt{34} \cdot 5}}\] Now, calculate \(\theta\): \[\theta \approx 127.31掳\] Therefore, the angle between the vectors (3,-5) and (-4,3) is approximately 127.31掳.

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