Question

Given a rectangular solid with sides of lengths \(1,1,\) and \(\sqrt{2}\), find the angle between a diagonal and one of the longest sides.

Short answer

The angle between the diagonal and the longest side is \(45^\circ\) or \(\frac{\pi}{4}\) radians.

Step-by-step solution

Identify the Diagonal of the Rectangular Solid

The diagonal of the rectangular solid can be determined using the Pythagorean theorem in three dimensions. Denote the sides of the rectangular solid as lengths 1, 1, and \(\sqrt{2}\). The diagonal \(d\) can be found by calculating the square root of the sum of the squares of the side lengths.\[d = \sqrt{1^2 + 1^2 + (\sqrt{2})^2} = \sqrt{1 + 1 + 2} = \sqrt{4} = 2\]Thus, the diagonal of the rectangular solid is 2.

Find the Unit Vector Along the Diagonal

The coordinates of the endpoint of the diagonal, starting from (0,0,0), are (1,1,\(\sqrt{2}\)). Therefore, the vector \(\mathbf{d}\) along the diagonal is \((1,1,\sqrt{2})\). The unit vector \(\mathbf{\hat{d}}\) in the direction of the diagonal is found by dividing each component by the magnitude of the diagonal:\[\mathbf{\hat{d}} = \left(\frac{1}{2}, \frac{1}{2}, \frac{\sqrt{2}}{2}\right)\]

Determine the Unit Vector Along the Longest Side

The longest side of lengths provided is \(\sqrt{2}\). This side is parallel to the z-axis. Therefore, the vector along the longest side is \((0,0,\sqrt{2})\) and its unit vector \(\mathbf{\hat{a}}\) is calculated as:\[\mathbf{\hat{a}} = \left(0, 0, 1\right)\]

Calculate the Dot Product of the Two Vectors

To find the angle between the diagonal and the longest side, calculate the dot product of their unit vectors.\[\mathbf{\hat{d}} \cdot \mathbf{\hat{a}} = \left(\frac{1}{2}\right)\cdot 0 + \left(\frac{1}{2}\right)\cdot 0 + \left(\frac{\sqrt{2}}{2}\right) \cdot 1 = \frac{\sqrt{2}}{2}\]

Use Dot Product to Find the Angle

Utilize the dot product formula that relates to the cosine of the angle \(\theta\) between two vectors, \(\mathbf{A}\) and \(\mathbf{B}\):\[\cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{\|\mathbf{A}\| \|\mathbf{B}\|}\]Since \(\|\mathbf{\hat{d}}\| = \|\mathbf{\hat{a}}\| = 1\), the expression simplifies to:\[\cos \theta = \mathbf{\hat{d}} \cdot \mathbf{\hat{a}} = \frac{\sqrt{2}}{2}\]This implies \(\theta = \cos^{-1}\left(\frac{\sqrt{2}}{2}\right)\), which is a standard angle.

Determine the Angle

The value \(\cos^{-1}\left(\frac{\sqrt{2}}{2}\right)\) corresponds to an angle \(\theta\) of \(45^\circ\) or \(\frac{\pi}{4}\) radians because \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).

Key concepts

3D geometry

Three-dimensional geometry is all about understanding shapes and objects that exist in a space with depth. While 2D geometry only deals with shapes on a plane, like squares and circles, 3D geometry includes objects like cubes, spheres, and rectangular solids.

In this exercise, we are dealing with a rectangular solid. Imagine a shoebox, with all right angles and three different lengths: length, width, and height. Here, we have sides of lengths 1, 1, and \(\sqrt{2}\). By knowing these, we can explore various properties, such as finding the diagonal within the solid.

To find a diagonal in 3D, we extend the Pythagorean theorem, usually used in 2D. Instead of squaring two side lengths, we now square all three to find the diagonal across the solid. This gives us a broader understanding of the shape's internal dimensions and helps us solve different geometric problems, like finding the angle between this diagonal and one of its sides.

dot product

The dot product is a crucial tool in vector mathematics, serving as a bridge to find the angle between two vectors. Vectors, which have both direction and magnitude, are essential in describing various elements in 3D geometry.

For the dot product, envision two vectors, \(\mathbf{A}\) and \(\mathbf{B}\). The calculation is straightforward: multiply the corresponding components and sum them up. Mathematically, it’s expressed as \(\mathbf{A} \cdot \mathbf{B} = A_1 B_1 + A_2 B_2 + A_3 B_3\).

In our example, we calculated a dot product to find the angle between the diagonal of the rectangular solid and one of its longest sides. Through the formula \(\cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{\|\mathbf{A}\| \|\mathbf{B}\|}\), we linked the dot product to the cosine of the angle \(\theta\). Because the vectors were unit vectors, their magnitudes were \(1\), simplifying our calculations. Hence, the dot product directly told us the cosine of our required angle.

unit vectors

Unit vectors simplify many calculations in vector mathematics as they have a magnitude of exactly 1. Think of them as setting a standard for direction, without affecting the vectors' direction. For example, a unit vector along the x-axis would be \((1, 0, 0)\), representing direction with no scaling effect.

In our problem, we used unit vectors to address both the diagonal and the longest side of the rectangular solid. For instance, the diagonal’s unit vector was \(\mathbf{\hat{d}} = \left(\frac{1}{2}, \frac{1}{2}, \frac{\sqrt{2}}{2}\right)\), a standardized direction of the diagonal division.

Unit vectors allow us to focus on the "shape" or direction in vector operations. When calculating angles, they’re crucial because their magnitude is neutralized. This characteristic makes them perfect for finding angles, as seen with the diagonal’s and longest side’s unit vectors. By using these vectors in the dot product, we found the cosine of the angle easily, leading to our solution.