Problem 52 - Precalculus with Limits

Question

Using the Angle Between Two Vectors Exercises \(49-52,\) find \(u \cdot v,\) where \(\theta\) is the angle between \(\mathbf{u}\) and \(v .\) $$\|\mathbf{u}\|=4,\|\mathbf{v}\|=12, \theta=\frac{\pi}{3}$$

Step-by-Step Solution

Answer

The dot product of vectors u and v is 24.

Step 1: Identify the Given Values

In the problem, we are given the magnitudes of the vectors u and v, and the angle between them: \(\|u\|\) = 4, \(\|v\|\) = 12, and \(\theta\) = \(\pi/3\).

Step 2: Apply the Dot Product Formula

Substitute the given values into the formula for the dot product: \(u \cdot v = \|u\|\|v\|cos(\theta)\). This gives us: \(u \cdot v = 4 * 12 * cos(\pi/3)\).

Step 3: Evaluate the Expression

Evaluate the expression for the dot product. \(cos(\pi/3) = 1/2\), so \(u \cdot v = 4 * 12 * 1/2\).

Step 4: Simplify the Result

Simplifying the above expression will give us the final answer: \(u \cdot v = 24\).

Key Concepts

Magnitude of a Vector

The magnitude of a vector is essentially its length. When we talk about the magnitude, think about how long a line is from its start to its finish. In a physical sense, for example, it could represent things like speed or force. Mathematically, the magnitude of a vector \( \mathbf{v} \) is denoted as \( \|\mathbf{v}\| \). When given a vector in three-dimensional space, say \( \mathbf{v} = (x, y, z) \), its magnitude is calculated by the formula: \[ \|\mathbf{v}\| = \sqrt{x^2 + y^2 + z^2} \]For our exercise, we are given two vectors with their magnitudes already specified: \( \|\mathbf{u}\| = 4 \) and \( \|\mathbf{v}\| = 12 \). Basically, this data tells us how long each of these vectors are, simplifying the process when we need to compute operations like the dot product.

Angle Between Vectors

The angle between two vectors is a measure of how "tilted" one vector is compared to the other. It is an indicator of their direction relative to each other. To find this angle, we can use the dot product formula. The formula relates the dot product with the magnitudes of the vectors and the cosine of the angle between them:\[ u \cdot v = \|u\| \|v\| \cos(\theta) \] Here, \( \theta \) represents the angle between vectors \( \mathbf{u} \) and \( \mathbf{v} \). In our exercise, we are given that \( \theta = \frac{\pi}{3} \). This is essentially 60 degrees, which is a common angle in trigonometry leading to simplifications when calculating with cosine.

Trigonometric Functions

Trigonometric functions help us relate angles to ratios of sides in right triangles. Among these, cosine (often written as \( \cos \)) is crucial when discussing vectors, especially when dealing with the dot product. The cosine of an angle in a triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. In our example, we find \( \cos(\theta) \), where \( \theta = \frac{\pi}{3} \). By looking at special triangles or using the unit circle, it's known that:\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]This result simplifies the calculation of the dot product significantly, as we see in our step-by-step solution. Such simplifications are invaluable in vectors and can be visualized easily with the help of trigonometric functions to understand the angle relations.