Find The Angle Between $u=i+\sqrt{7} J$ And $v=-i+9 J$. Round To The Nearest Tenth Of A Degree.a. 68.9 ∘ 68.9^\circ 68. 9 ∘ B. 27.0 ∘ 27.0^\circ 27. 0 ∘ C. 14.4 ∘ 14.4^\circ 14. 4 ∘ D. 14.5 ∘ 14.5^\circ 14. 5 ∘ Please Select The Best Answer

Introduction

In mathematics, vectors are used to represent quantities with both magnitude and direction. The angle between two vectors is an essential concept in vector calculus, and it has numerous applications in physics, engineering, and other fields. In this article, we will discuss how to find the angle between two vectors using the dot product formula.

The Dot Product Formula

The dot product formula is used to find the angle between two vectors. Given two vectors $\mathbf{u}$ and $\mathbf{v}$, the dot product formula is:

$$\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta$$

where $|\mathbf{u}|$ and $|\mathbf{v}|$ are the magnitudes of the vectors, and $\theta$ is the angle between them.

Finding the Magnitudes of the Vectors

To find the angle between two vectors, we need to find their magnitudes first. The magnitude of a vector $\mathbf{u} = a \mathbf{i} + b \mathbf{j}$ is given by:

$$|\mathbf{u}| = \sqrt{a^2 + b^2}$$

Let's find the magnitudes of the vectors $\mathbf{u} = i + \sqrt{7} j$ and $\mathbf{v} = -i + 9 j$.

Magnitude of Vector $\mathbf{u}$

The magnitude of vector $\mathbf{u}$ is:

$$|\mathbf{u}| = \sqrt{1^2 + (\sqrt{7})^2} = \sqrt{1 + 7} = \sqrt{8}$$

Magnitude of Vector $\mathbf{v}$

The magnitude of vector $\mathbf{v}$ is:

$$|\mathbf{v}| = \sqrt{(-1)^2 + 9^2} = \sqrt{1 + 81} = \sqrt{82}$$

Finding the Dot Product

Now that we have the magnitudes of the vectors, we can find the dot product using the formula:

$$\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta$$

We need to find the dot product of the vectors $\mathbf{u}$ and $\mathbf{v}$.

Dot Product of Vectors $\mathbf{u}$ and $\mathbf{v}$

The dot product of vectors $\mathbf{u}$ and $\mathbf{v}$ is:

$$\mathbf{u} \cdot \mathbf{v} = (i + \sqrt{7} j) \cdot (-i + 9 j)$$

Using the distributive property, we get:

$$\mathbf{u} \cdot \mathbf{v} = -1 + 9 \sqrt{7}$$

Finding the Angle

Now that we have the dot product, we can find the angle between the vectors using the formula:

$$\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|}$$

We can plug in the values we found earlier:

$$\cos \theta = \frac{-1 + 9 \sqrt{7}}{\sqrt{8} \sqrt{82}}$$

Simplifying the expression, we get:

$$\cos \theta = \frac{-1 + 9 \sqrt{7}}{\sqrt{656}}$$

Now, we can use a calculator to find the value of $\cos \theta$.

Value of $\cos \theta$

Using a calculator, we get:

$$\cos \theta \approx 0.9993$$

Finding the Angle in Degrees

Now that we have the value of $\cos \theta$, we can find the angle in degrees using the inverse cosine function.

Angle in Degrees

Using a calculator, we get:

$$\theta \approx \cos^{-1} (0.9993) \approx 14.5^\circ$$

Conclusion

In this article, we discussed how to find the angle between two vectors using the dot product formula. We found the magnitudes of the vectors, the dot product, and the angle in degrees. The final answer is:

• 14.5^\circ

This is the correct answer. The other options are incorrect.

References

• [1] "Vector Calculus" by Michael Corral, 2018.
• [2] "Calculus" by Michael Spivak, 2008.

Note

Frequently Asked Questions

In this article, we will answer some frequently asked questions about finding the angle between two vectors.

Q: What is the dot product formula?

A: The dot product formula is used to find the angle between two vectors. Given two vectors $\mathbf{u}$ and $\mathbf{v}$, the dot product formula is:

$$\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta$$

where $|\mathbf{u}|$ and $|\mathbf{v}|$ are the magnitudes of the vectors, and $\theta$ is the angle between them.

Q: How do I find the magnitudes of the vectors?

A: To find the magnitudes of the vectors, you need to use the formula:

$$|\mathbf{u}| = \sqrt{a^2 + b^2}$$

where $a$ and $b$ are the components of the vector.

Q: What is the difference between the dot product and the cross product?

A: The dot product and the cross product are both used to find the relationship between two vectors, but they are used for different purposes. The dot product is used to find the angle between two vectors, while the cross product is used to find the area of the parallelogram formed by the two vectors.

Q: Can I use the dot product formula to find the angle between two vectors in 3D space?

A: Yes, you can use the dot product formula to find the angle between two vectors in 3D space. However, you need to use the formula:

$$\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta$$

where $\mathbf{u}$ and $\mathbf{v}$ are the vectors in 3D space.

Q: How do I find the angle between two vectors in degrees?

A: To find the angle between two vectors in degrees, you need to use the inverse cosine function:

$$\theta = \cos^{-1} \left( \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|} \right)$$

Q: What is the unit of measurement for the angle between two vectors?

A: The unit of measurement for the angle between two vectors is degrees.

Q: Can I use the dot product formula to find the angle between two vectors with complex components?

A: Yes, you can use the dot product formula to find the angle between two vectors with complex components. However, you need to use the formula:

$$\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta$$

where $\mathbf{u}$ and $\mathbf{v}$ are the vectors with complex components.

Q: How do I find the angle between two vectors with different units of measurement?

A: To find the angle between two vectors with different units of measurement, you need to convert the units of measurement to a common unit. Then, you can use the dot product formula to find the angle between the vectors.

Conclusion

In this article, we answered some frequently asked questions about finding the angle between two vectors. We discussed the dot product formula, the magnitudes of the vectors, the difference between the dot product and the cross product, and how to find the angle between two vectors in 3D space. We also discussed how to find the angle between two vectors with complex components and how to find the angle between two vectors with different units of measurement.