Problem 50 - Precalculus with Limits

Question

In Exercises 49-52, find \(\mathbf{u \cdot v}\), where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\). \(||\mathbf{u}|| = 100\), \(||\mathbf{v}|| = 250\), \(\theta = \dfrac{\pi}{6}\)

Step-by-Step Solution

Answer

The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is \(12500\sqrt{3}\).

Step 1: Calculate the cos of the angle

As per the given condition, the angle \(\theta = \dfrac{\pi}{6}\). The cos of this angle needs to be calculated as it is a part of the formula for the dot product. So, calculate the cos(\(\dfrac{\pi}{6}\)) which results in \(\sqrt{3}/2\).

Step 2: Apply the dot product formula

The formula for the dot product of two vectors is given by \(\mathbf{u \cdot v} = ||\mathbf{u}|| \cdot ||\mathbf{v}|| \cdot \cos(\theta)\). Apply this formula, substituting the given vector magnitudes and the calculated cos of the angle. So, it becomes \(\mathbf{u \cdot v} = 100 \cdot 250 \cdot \sqrt{3}/2\).

Step 3: Compute the value

On calculating, the value of \(\mathbf{u \cdot v}\) becomes \(12500\sqrt{3}\).

Key Concepts

Vector Operations

Understanding vector operations is fundamental in fields such as physics, engineering, and computer science. Vectors are geometric entities that have both magnitude and direction. When operating on vectors, common operations include addition, subtraction, scaling (multiplying by a scalar), and the dot product.

The dot product, also known as the scalar product, is particularly noteworthy. It takes two equal-length sequences of numbers (usually coordinates), and returns a single number. This operation combines two vectors and yields information about their directional relationship. Specifically, the dot product can tell us if two vectors are orthogonal (perpendicular), pointing in the same or opposite directions, or something in between. It is given by the formula \( \mathbf{u \cdot v} = ||\mathbf{u}|| \cdot ||\mathbf{v}|| \cdot \cos(\theta) \), where \(||\mathbf{u}||\) and \(||\mathbf{v}||\) are the magnitudes of vectors \(\mathbf{u}\) and \(\mathbf{v}\), and \(\theta\) is the angle between them.

Cosine of an Angle

The cosine of an angle is a trigonometric function that is extremely valuable when working with right triangles and vector operations. In the context of the dot product, the cosine helps determine the extent to which two vectors share the same direction.

For any angle \(\theta\), the cosine function returns a value between -1 and 1. A cosine of 1 means the vectors are pointed in exactly the same direction, -1 implies they are opposite, and a cosine of 0 indicates that the vectors are perpendicular. When calculating the dot product, it's crucial to accurately determine the cosine of the included angle \(\theta\). For angles commonly found in problems, such as \(\dfrac{\pi}{6}\), it's handy to know their cosine values or how to look them up—\(\sqrt{3}/2\) in this case.

Magnitude of a Vector

The magnitude of a vector represents its length or size and is a measure of how long the vector is in the space it occupies. It is denoted by two vertical lines on either side of the vector, like \(||\mathbf{u}||\) for the magnitude of vector \(\mathbf{u}\). The magnitude is a scalar quantity, meaning it has no direction, just size.

Calculating the magnitude of a vector depends on its components. For a two-dimensional vector \(\mathbf{u} = (u_1, u_2)\), the magnitude is found using the Pythagorean theorem: \(||\mathbf{u}|| = \sqrt{u_1^2 + u_2^2}\). For a three-dimensional vector \(\mathbf{u} = (u_1, u_2, u_3)\), it is \(||\mathbf{u}|| = \sqrt{u_1^2 + u_2^2 + u_3^2}\). Knowing how to compute magnitudes is crucial for vector operations, especially when calculating the dot product.