Question

In Exercises 7 and \(8,\) find \(u \cdot v\). \(\|\mathbf{u}\|=8,\|\mathbf{v}\|=5,\) and the angle between \(\mathbf{u}\) and \(\mathbf{v}\) is \(\pi / 3\).

Short answer

The dot product of vectors \(u\) and \(v\) is 20.

Step-by-step solution

Identify the magnitudes and angle

Here, the magnitude of vector \(u\) is 8 (\(\|u\|=8\)) and the magnitude of vector \(v\) is 5 (\(\|v\|=5\)). The angle between the vectors (\(\theta\)) is \(\pi / 3\).

Use the dot product formula

Apply the values into the given formula for dot product: \(u \cdot v = \|u\| * \|v\| * \cos(\theta)\). This gives \(u \cdot v = 8 * 5 * \cos(\pi / 3)\).

Calculation

Solve the equation. Taking the cosine of \(\pi / 3\) gives 0.5. Therefore, the solution is \(u \cdot v = 8 * 5 * 0.5 = 20\).

Key concepts

Vector Magnitude

When we talk about the magnitude of a vector, we are referring to its length. Think of it as the distance from the starting point (or tail) of the arrow to the tip of the arrow that represents our vector. For those more mathematically inclined, the magnitude of a vector \textbf{v} is denoted as \(\|\mathbf{v}\|\) and can be calculated using the Pythagorean theorem if its components along the axes are known, resulting in the square root of the sum of the squares of its components.

For example, if a vector \(\mathbf{v}\) has components \(v_x\) and \(v_y\) in a 2-dimensional space, the magnitude is obtained by \(\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2}\). The same concept applies to 3-dimensional vectors, just with an additional component added to the formula. Understanding vector magnitude is critical since it's used in numerous vector operations, including dot product, which requires both the magnitudes of the involved vectors and the angle between them.

Angle Between Vectors

The angle between two vectors is a measure of how they're directed with respect to each other. It's applicable in various branches of mathematics and physics, including vector algebra and mechanics. To find the angle between vectors, we often use trigonometric functions – the cosine function being one of the most relevant in the case of the dot product.

To conceptualize this, imagine two vectors originating from the same point and spreading out. The angle between them is the figure formed at the point where they meet. The earlier you grab this concept, the easier it becomes to work with vector operations. In our exercise, the angle \(\theta\) is given as \(\pi / 3\), which is familiar ground if you're versed in radians. It's crucial to consider that the dot product of two vectors is maximal when they point in the same direction (the angle is 0) and minimal when they point opposite to each other (the angle is \(\pi\)), assuming their magnitudes are constant.

Cosine Function

The cosine function is a trigonometric function that relates the angles of a right triangle to the lengths of its sides. As the angle \(\theta\) changes, so does the cosine value, which ranges between -1 and 1. In the context of vectors, the cosine of the angle between two vectors is used in calculating the dot product.

In our exercise, we calculate \(\cos(\pi / 3)\), which results in 0.5. This particular cosine value is very crucial because it shows up often in problems involving equilateral triangles or hexagonal symmetry, where the internal angles are all \(\pi / 3\). Memorize these key cosine values to save time in calculations.

Importance in Dot Product

The dot product itself is actually a projection of one vector onto another, and the cosine function tells us the extent of overlap or agreement in direction between the vectors. We multiply the cosines of the angle with the magnitudes of the vectors to derive the dot product, showing the intertwined relationship between the angle's cosine and the magnitudes of vectors.