Problem 2 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

$$ |\overrightarrow{\mathbf{u}}|=5,|\overrightarrow{\mathbf{v}}|=7 \text { , and the angle between } \overrightarrow{\mathbf{u}} \text { and } \overrightarrow{\mathbf{v}} \text { is } \frac{\pi}{6} \text { . } $$

Step-by-Step Solution

Answer

The dot product of \(\overrightarrow{\mathbf{u}}\) and \(\overrightarrow{\mathbf{v}}\) is \(\frac{35\sqrt{3}}{2}\)

Step 1: Understand the Dot Product Formula

The formula for the dot product of two vectors in terms of their magnitudes and the angle between them is given by: \(\overrightarrow{\mathbf{u}} \cdot \overrightarrow{\mathbf{v}} = |\overrightarrow{\mathbf{u}}| |\overrightarrow{\mathbf{v}}| \cos(\theta)\) where \(\overrightarrow{\mathbf{u}}\) and \(\overrightarrow{\mathbf{v}}\) are the vectors, \(|\overrightarrow{\mathbf{u}}|\) and \(|\overrightarrow{\mathbf{v}}|\) their magnitudes, and \(\theta\) the angle between them.

Step 2: Substitute the Given Values

Substitute the given values into the formula: \(|\overrightarrow{\mathbf{u}} \cdot \overrightarrow{\mathbf{v}}| = |5| |7| \cos\left(\frac{\pi}{6}\right)\)

Step 3: Calculate the Dot Product

Calculate the dot product: \(|\overrightarrow{\mathbf{u}} \cdot \overrightarrow{\mathbf{v}}| = 5 \times 7 \times \frac{\sqrt{3}}{2} = \frac{35\sqrt{3}}{2}\)

Key Concepts

Understanding Vector Magnitudes

The magnitude of a vector is a measure of its length or size. Imagine drawing an arrow on a piece of paper; the length of that arrow represents the vector's magnitude. In mathematical terms, if we consider a vector \( \vec{v} \) with components \( (x, y) \) in two-dimensional space, the magnitude—denoted as \( |\/vec{v}| \)—is calculated using the Pythagorean theorem: \(|\vec{v}| = \sqrt{x^2 + y^2}\).

For vectors in three-dimensional space, the formula extends to \(|\vec{v}| = \sqrt{x^2 + y^2 + z^2}\). The magnitudes are always non-negative and provide crucial information when working with vectors in physics and engineering, as they relate to concepts like force, velocity, and displacement.

Deciphering the Angle Between Vectors

The angle between two vectors is a fundamental concept when determining their relational direction to one another. In a two-dimensional plane, if you place the tails of two vectors together, the angle they form is the angle we're interested in. This angle is crucial for many applications, such as when calculating torque or understanding the orientation of objects in space.

To find the angle between two vectors \( \vec{u} \) and \( \vec{v} \) mathematically, we use trigonometry and the dot product formula. However, it's also important to note that the angle between two vectors is confined to a range of \( 0 \) to \( \pi \) radians (or \( 0 \) to \( 180 \) degrees), and it represents the smallest angle needed to 'rotate' one vector onto the other.

Grasping the Cosine of Angle in Dot Product

In the dot product formula \(|\vec{u}| |\vec{v}| \cos(\theta)\), the cosine of the angle \( \theta \) between the vectors is the key to linking their magnitudes with their directional relationship. The cosine function returns a scalar value that adjusts the product of the vectors' magnitudes according to how aligned, or opposite, their directions are.

A cosine of \( 1 \) means the vectors are pointing in the exact same direction (the angle is \( 0 \) radians), while a cosine of \( -1 \) indicates they are opposite (the angle is \( \pi \) radians). A cosine of \( 0 \) means the vectors are perpendicular to each other. Thus, by using the cosine in the dot product, we can quantify the extent to which two vectors are pointing in the same direction.