Problem 37 - Precalculus with Limits

Question

Finding the Angle Between Two Vectors In Exercises \(31-40,\) find the angle \(\theta\) between the vectors. $$\begin{aligned} \mathbf{u} &=5 \mathbf{i}+5 \mathbf{j} \\ \mathbf{v} &=-6 \mathbf{i}+6 \mathbf{j} \end{aligned}$$

Step-by-Step Solution

Answer

So, the angle between vector \(\mathbf{u}\) and vector \(\mathbf{v}\) is 90 degrees or \(\pi / 2\) radians.

Step 1: Find the dot product of vectors u and v

The dot product is defined as \(\mathbf{u} \cdot \mathbf{v} = \sum{u_iv_i}\). So with \(\mathbf{u} = 5\mathbf{i} + 5\mathbf{j}\) and \(\mathbf{v} = -6\mathbf{i} + 6\mathbf{j}\), we get \(\mathbf{u} \cdot \mathbf{v} = (5*-6) + (5*6) = 0\)

Step 2: Compute the magnitudes of vectors u and v

The magnitude of a vector is the square root of the sum of the squares of its components. For \(\mathbf{u}\), we compute it as \(||\mathbf{u}|| = \sqrt{(5^2) + (5^2)} = 5\sqrt{2}\). Similarly, for \(\mathbf{v}\), we compute \(||\mathbf{v}|| = \sqrt{(-6^2) + (6^2)} = 6\sqrt{2}\)

Step 3: Apply the dot product formula

The dot product expression for the cosine of the angle between two vectors is cos\(\theta\) = (u • v) / (||u|| ||v||). Substituting the results we calculated, we get cos\(\theta\) = (0) / (5\sqrt{2} * 6\sqrt{2}). Simplifying this gives cos\(\theta\) = 0. The angle that has a cosine of 0 is 90 degrees or \(\pi / 2\) radians

Key Concepts

Dot Product

When we talk about the dot product—also known as the scalar product—we refer to an operation that combines two vectors to produce a scalar value. It doesn't provide a vector like the cross product does, but a simple number that tells us something about the relationship between the two vectors.

The dot product is central in finding the angle between two vectors because it measures the extent to which two vectors point in the same direction. It is determined by multiplying the corresponding components of the vectors and then summing up these products. Expressed algebraically, if you have two vectors, \(\mathbf{u} = u_1\mathbf{i} + u_2\mathbf{j}\) and \(\mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j}\), then the dot product \(\mathbf{u} \cdot \mathbf{v}\) is given by the formula \(u_1v_1 + u_2v_2\).

If the dot product is positive, the vectors are pointing in somewhat the same direction. If it's zero—as in the given exercise—the vectors are perpendicular to each other, and if the dot product is negative, the vectors are pointing in opposite directions.

This concept becomes even more intriguing as it connects to other vector properties like magnitude and the angle between them, which are further explained in the next sections.

Vector Magnitude

Understanding the magnitude of a vector is key to many vector-related operations, such as determining length, comparing vector sizes, or normalizing vectors. The magnitude, also referred to as the 'length' or 'size' of a vector, provides the quantitative measure of how long the vector is.

The magnitude is essentially the distance from the origin of the vector to its tip when it's plotted on a coordinate system. To calculate the magnitude of a vector, you take the square root of the sum of the squares of its components. For a two-dimensional vector \( \mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j}\), the magnitude is \( ||\mathbf{v}|| = \sqrt{v_1^2 + v_2^2} \). In the three-dimensional case, you would also add the square of the third component.

Calculating Magnitude

In our exercise, for vector \(\mathbf{u}\), the magnitude calculation is \(||\mathbf{u}|| = \sqrt{5^2 + 5^2} = 5\sqrt{2}\), which means the 'length' of vector \(\mathbf{u}\) is \(5\sqrt{2}\). This gives us a clear measurement that we can use in comparison with other vectors, or to calculate other properties, such as the angle between two vectors.

Cosine Formula for Vectors

The cosine formula, as applied to vectors, is a powerful tool for finding the angle between two non-zero vectors. This formula relies on both the dot product and the magnitudes of the vectors involved. The underlying principle is that the cosine of the angle \(\theta\) between two vectors can be found by dividing their dot product by the product of their magnitudes.

The formula looks like this: \(\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| ||\mathbf{v}||}\). What this expression is really saying is that the cosine of the angle is proportional to how much the vectors are pointing in the same direction, normalized by the size of the vectors themselves. The result of this operation is a dimensionless number between -1 and 1, corresponding to the cosine of the angle between the vectors.

Applying the Cosine Formula

In the exercise provided, the cosine formula is used in the latter part of the solution. After calculating the dot product, which turns out to be 0, and determining the magnitudes of each vector, we apply these into the cosine formula. Since the dot product is 0, the formula simplifies to \(\cos(\theta) = 0\), which indicates the vectors are orthogonal. Hence, the angle between them is 90 degrees or \(\frac{\pi}{2}\) radians. The cosine formula is usually the last step in calculating the angle between two vectors once the dot product and magnitudes are known.