Find the angle between the vectors a = (-1, 6, 7) and b = (5, 3, 1). First find an exact expression and then approximate to the nearest degree.
Understand the Problem
The question is asking to find the angle between two vectors by first calculating an exact expression for the angle and then approximating it to the nearest degree.
Answer
$ \theta = \cos^{-1}\left(\frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| \cdot |\mathbf{B}|}\right) $
Answer for screen readers
The final answer will depend on the specific values of the vectors provided. For example, if we have $\mathbf{A} = (3, 4)$ and $\mathbf{B} = (4, 3)$, we would follow through with these calculations to determine the angle between these two vectors.
Steps to Solve
- Define the vectors
Let the two vectors be $\mathbf{A}$ and $\mathbf{B}$. For example, let $\mathbf{A} = (x_1, y_1)$ and $\mathbf{B} = (x_2, y_2)$.
- Calculate the dot product
The dot product of the two vectors can be calculated using the formula: $$ \mathbf{A} \cdot \mathbf{B} = x_1 \cdot x_2 + y_1 \cdot y_2 $$
- Calculate the magnitudes of the vectors
The magnitude of vector $\mathbf{A}$ is: $$ |\mathbf{A}| = \sqrt{x_1^2 + y_1^2} $$
The magnitude of vector $\mathbf{B}$ is: $$ |\mathbf{B}| = \sqrt{x_2^2 + y_2^2} $$
- Use the cosine formula to find the angle
The cosine of the angle $\theta$ between the two vectors is given by: $$ \cos(\theta) = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| \cdot |\mathbf{B}|} $$
- Calculate the angle
To find the angle $\theta$, take the inverse cosine: $$ \theta = \cos^{-1}\left(\frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| \cdot |\mathbf{B}|}\right) $$
- Approximate the angle
Finally, round the angle $\theta$ to the nearest degree.
The final answer will depend on the specific values of the vectors provided. For example, if we have $\mathbf{A} = (3, 4)$ and $\mathbf{B} = (4, 3)$, we would follow through with these calculations to determine the angle between these two vectors.
More Information
The angle between vectors is useful in various applications such as physics, computer graphics, and machine learning. Calculating the angle can help determine the direction and relationship of vectors in multi-dimensional space.
Tips
- Forgetting to take the square root when calculating magnitudes.
- Confusing the dot product formula.
- Not using the correct range for the inverse cosine function, leading to errors in angle values.
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