Problem 84 - Precalculus

Question

The definition of a dot product and the formula to find the angle between two vectors can be extended and applied to vectors with more than two components. A rectangular box has sides with lengths 12 feet, 7 feet, and 9 feet. Find the angle, to the nearest degree, between the diagonal and the side with length 7 feet.

Step-by-Step Solution

Answer

The angle between the diagonal and the 7-feet side is approximately 62 degrees.

Step 1: Identify Vectors

First, we need to identify the vectors involved. Let the vector along the diagonal of the rectangle box be \( \mathbf{d} = (12, 7, 9) \). The vector along the side with length 7 feet is \( \mathbf{s} = (0, 7, 0) \), as it lies along the y-axis.

Step 2: Calculate the Magnitude of Vectors

Calculate the magnitudes of both vectors. The magnitude of \( \mathbf{d} \) is given by \[ \| \mathbf{d} \| = \sqrt{12^2 + 7^2 + 9^2} = \sqrt{244}. \] Similarly, the magnitude of \( \mathbf{s} \) is \[ \| \mathbf{s} \| = \sqrt{0^2 + 7^2 + 0^2} = 7. \]

Step 3: Find Dot Product

Compute the dot product of \( \mathbf{d} \) and \( \mathbf{s} \). \[ \mathbf{d} \cdot \mathbf{s} = (12 \times 0) + (7 \times 7) + (9 \times 0) = 49. \]

Step 4: Use Dot Product to Find Angle

Use the dot product to find the cosine of the angle \( \theta \) between the vectors using the formula: \[ \cos \theta = \frac{\mathbf{d} \cdot \mathbf{s}}{\|\mathbf{d}\| \|\mathbf{s}\|}. \] Substitute the known values: \[ \cos \theta = \frac{49}{\sqrt{244} \times 7}. \]

Step 5: Calculate \( \theta \)

Solve for \( \theta \) by taking the inverse cosine (arccos) of \( \cos \theta \): \[ \theta = \cos^{-1}\left(\frac{49}{\sqrt{244} \times 7}\right). \] This will give \( \theta \approx 62.3^\circ \) rounding it to the nearest degree results in \( 62^\circ \).

Key Concepts

Angle Between Vectors

Understanding the angle between vectors is crucial in various fields, including physics and engineering. The angle between two vectors provides insight into how aligned these vectors are in space. Particularly, when two vectors are parallel, their angle is 0 degrees, and when they are orthogonal (perpendicular), the angle is 90 degrees. This concept is calculated using the dot product formula:\[\cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{\|\mathbf{A}\| \|\mathbf{B}\|}\]where \(\theta\) is the angle between vectors \(\mathbf{A}\) and \(\mathbf{B}\). The dot product \(\mathbf{A} \cdot \mathbf{B}\) measures the extent to which two vectors point in the same direction. If the dot product is positive, the angle is less than 90 degrees, indicating vectors point in the same general direction.- **Steps to find angle between vectors:** - Calculate the magnitudes of each vector. - Find the dot product of the vectors. - Use the formula to solve for the angle.In our exercise, calculating the angle between the diagonal of a rectangular box and one of its sides uses these principles. By substituting the vectors and their magnitudes into the formula, we can find the angle efficiently.

Magnitude of Vectors

The magnitude of a vector, often referred to as the length or size, is a measure of its extent in space. Calculating the magnitude involves applying the Pythagorean theorem in multidimensional space. For example, a vector \( \mathbf{v} = (x, y, z) \) has a magnitude calculated as:\[\| \mathbf{v} \| = \sqrt{x^2 + y^2 + z^2}\]The magnitude is always a non-negative number and gives an essential geometric interpretation of the vector itself. Knowing the magnitude is important, especially when calculating the angle between vectors or performing vector operations like normalizing a vector.- **Tips for calculating magnitude:** - Square each component. - Sum the squared components. - Take the square root of the sum.In the context of the exercise, finding the magnitudes of both the diagonal vector and the side vector were necessary steps. The diagonal’s magnitude involved using its three-dimensional components, while the side vector, lying along the y-axis, only required a simple calculation.

Rectangular Box Diagonal

The diagonal of a rectangular box, also called the space diagonal, is a line segment that traverses from one corner of the box to the opposite corner. Calculating its vector involves using the box’s dimensions as vector components. In our context, if a box has dimensions \(a\), \(b\), and \(c\), the vector representing its diagonal is \((a, b, c)\).- **Steps to find a diagonal:** - Use the lengths of the box as the vector’s components. - Consider it as a three-dimensional vector. - Its magnitude can be computed, which also yields the length of the diagonal.In our problem, the vector representing the diagonal was \( (12, 7, 9) \). Understanding how to represent and calculate the diagonal is fundamental in solving this problem since it directly ties into finding the angle between the diagonal and a side of the box. The diagonal not only defines a spatial path through the box but also serves as a key component in determining the relative angle with each of the box's sides.