Problem 50 - 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 \(v .\) $$\|\mathbf{u}\|=100,\|\mathbf{v}\|=250, \quad \theta=\frac{\pi}{6}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the problem

The problem provides the magnitude of two vectors \(u\) and \(v\), and the angle between them, and asks for their dot product. The dot product of two vectors can be found using the magnitudes of the vectors and the cosine of the angle between them.

Step 2: Substitute the given values into the formula

We substitute the provided magnitudes for \(u\), \(v\), and the angle \(\theta\) into the dot product formula. So, \(u \cdot v = \|u\| \|v\| \cos(\theta)\), becomes \(u \cdot v = 100 \times 250 \times \cos\left(\frac{\pi}{6}\right)\).

Step 3: Calculate the dot product

After substituting, we multiply the magnitudes and the cosine of the angle to get the dot product. The cosine of \(\pi/6\) is \(\sqrt{3}/2\), so the expression simplifies to \(u \cdot v = 100 \times 250 \times \sqrt{3}/2\). This yields \(u \cdot v = 12500\sqrt{3}\).

Key Concepts

Angle Between Two Vectors

The angle between two vectors, often denoted by \( \theta \), plays a crucial role in understanding vector relationships in mathematics. It's the measure of rotation needed to align one vector with another. This angle can range from \( 0 \) to \( \pi \) radians (or \( 0 \) to \( 180 \) degrees).
Understanding this angle helps determine how "closely" vectors point towards each other. If \( \theta = 0 \), the vectors point in the same direction. If \( \theta = \pi \), they are directly opposite.

• To measure this angle, specific information like the vectors' magnitudes and their dot product are essential.
• Calculating the dot product involves these angled relationships and provides insight into the vectors' geometric interaction.

Magnitude of a Vector

The magnitude of a vector, denoted as \( \| \mathbf{v} \| \), is the length or size of the vector. It tells us how much ground the vector covers in space, which is fundamental in many applications like physics and engineering.
This is similar to finding the length of a line segment.

• For a vector \( \mathbf{v} = (a, b) \), the magnitude is calculated as \( \| \mathbf{v} \| = \sqrt{a^2 + b^2} \).
• In three dimensions, if \( \mathbf{v} = (a, b, c) \), then \( \| \mathbf{v} \| = \sqrt{a^2 + b^2 + c^2} \).
• The magnitude is always a non-negative number.

Knowing the magnitudes helps when using the dot product formula, which is critical for calculating vector interactions.

Cosine of an Angle

The cosine of an angle, often expressed as \( \cos(\theta) \), is a fundamental trigonometric function. It represents the x-coordinate of a point on the unit circle at a given angle \( \theta \). This concept extends beyond geometry and is pivotal in physics, particularly with vectors.

• The cosine value ranges from \(-1\) to \(1\).
• At \( \theta = 0 \), \( \cos(\theta) = 1 \), indicating complete alignment.
• At \( \theta = \pi \), \( \cos(\theta) = -1 \), signifying opposite directions.

For vector calculations, the cosine of the angle between two vectors is used to determine the dot product. This relationship \( u \cdot v = \|u\| \|v\| \cos(\theta) \) connects trigonometry with vector algebra in a meaningful way. In our exercise, with \( \theta = \pi/6 \), \( \cos(\pi/6) = \sqrt{3}/2 \). This simplifies the calculation of the dot product.