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Chapter 6: Problem 19

Find the angle between \(\mathbf{v}\) and \(\mathbf{w} .\) Round to the nearest tenth of a degree. $$\mathbf{v}=-3 \mathbf{i}+2 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}-\mathbf{j}$$

Short Answer

Expert verified
The angle between vectors \(\mathbf{v}\) and \(\mathbf{w}\) is approximately \(115.5\) degrees when rounded to the nearest tenth of a degree.

Step by step solution

01

Calculate the Dot Product

First, compute the dot product of \(\mathbf{v}\) and \(\mathbf{w}\). The dot product of two vectors is the sum of the products of their respective components. So, calculate \(\mathbf{v} \cdot \mathbf{w} = (-3)*(4) + (2)*(-1) = -14.\)
02

Calculate the Magnitudes

Next, calculate the magnitudes (lengths) of the two vectors \(\mathbf{v}\) and \(\mathbf{w}\). The magnitude of a vector is given by the square root of the sum of the squares of its components. Therefore, the magnitudes of vectors \(\mathbf{v}\) and \(\mathbf{w}\) are \(|\mathbf{v}| = \sqrt{(-3)^2 + 2^2} = \sqrt{13}\) and \(|\mathbf{w}| = \sqrt{4^2 + (-1)^2} = \sqrt{17}\), respectively.
03

Calculate the Angle

Finally, compute the angle \(\theta\) between \(\mathbf{v}\) and \(\mathbf{w}\) using the formula \(\cos \theta = \frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}| |\mathbf{w}|}\). Substituting the values, we get \(\cos \theta = \frac{-14}{\sqrt{13} * \sqrt{17}}\). To find the angle in degrees, apply the inverse cosine function and then convert the result from radians to degrees using the relation \(\1 rad = \frac{180}{\pi} deg\). So, \(\theta = \cos^{-1}\left(\frac{-14}{\sqrt{221}}\right) * \frac{180}{\pi}\). Round the result to the nearest tenth of a degree to get the final answer.

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Most popular questions from this chapter

Using words and no symbols, describe how to find the \(\mathrm{d}\) product of two vectors with the alternative formula $$\mathbf{v} \cdot \mathbf{w}=\|\mathbf{v}\|\|\mathbf{w}\| \cos \theta$$

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