Problem 80 - Precalculus with Limits

Question

In Exercises 79 and 80, use the Law of Cosines to find the angle \(\alpha\) between the vectors. ( Assume \(\small{0^{\circ} \neq \alpha \neq 190^{\circ}}\). ) \(\small{\mathbf{v} = \mathbf{i} + 2\mathbf{j}}\), \(\small{\mathbf{w} = 2\mathbf{i} - \mathbf{j}}\)

Step-by-Step Solution

Answer

So, the angle \(\alpha\) between the vectors is \(90^{\circ}\).

Step 1: Identifying the Vectors

The two given vectors are \(\mathbf{v} = \mathbf{i} + 2\mathbf{j}\) and \(\mathbf{w} = 2\mathbf{i} - \mathbf{j}\).

Step 2: Calculating Magnitudes

The magnitude of vector \(\mathbf{v}\) is given by \(\|\mathbf{v}\| = \sqrt{(1)^2 + (2)^2} = \sqrt{5}\), and the magnitude of vector \(\mathbf{w}\) is given by \(\|\mathbf{w}\| = \sqrt{(2)^2 + (-1)^2} = \sqrt{5}\).

Step 3: Calculating Dot Product

Dot product of vectors \(\mathbf{v}\) and \(\mathbf{w}\) is given by \(\mathbf{v} \cdot \mathbf{w} = (1)(2) + (2)(-1) = 0\).

Step 4: Solving for Angle

We use law of cosines which is \(\cos(\alpha) = \frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{v}\|\|\mathbf{w}\|}\). Substituting the found values we get \(\cos(\alpha) = \frac{0}{\sqrt{5} \cdot \sqrt{5}} = 0\). So, the angle \(\alpha\) is the \(\arccos(0) = 90^{°}\).

Key Concepts

Vector Magnitude

Understanding the magnitude of a vector is crucial when dealing with the Law of Cosines in relation to vectors. The magnitude, often referred to as the length, represents the size of the vector regardless of its direction. In mathematical terms, to calculate the magnitude of a two-dimensional vector \(\mathbf{v} = x\mathbf{i} + y\mathbf{j}\), we use the Pythagorean theorem: \(\|\mathbf{v}\| = \sqrt{x^2 + y^2}\). In simpler terms, think of this as the distance from the origin to the point \(x, y\) in a Cartesian plane. For instance, the magnitude of vector \(\mathbf{v} = \mathbf{i} + 2\mathbf{j}\) is \(\sqrt{1^2 + 2^2} = \sqrt{5}\).

It's essential to accurately compute the magnitude, as it's used in conjunction with the dot product to determine the angle between vectors. A common exercise improvement advice is to practice visualizing vectors and their magnitudes on a graph, which can help solidify the concept of vector magnitude and give a geometric interpretation to an otherwise abstract calculation.

Dot Product

The dot product is a way of multiplying two vectors to find a scalar quantity. It is an operation that combines two vectors and gives us information about their relationship, specifically their magnitude and the cosine of the angle between them. The formula for the dot product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) in two dimensions is \(\mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y\). In the given problem, we use this formula to find the dot product of vectors \(\mathbf{v}\) and \(\mathbf{w}\), which turns out to be zero. This indicates that the vectors are perpendicular, as the cosine of the angle between them is zero.

Keep in mind when studying the dot product that it is commutative, meaning \(\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}\), and it provides insight not only into the angle but also the projection of one vector onto another. By mastering the application of the dot product, one can solve various problems related to the orientation and interaction of vectors in a plane.

Angle Between Vectors

Determining the angle between vectors is a key part of vector analysis in mathematics and physics. When the dot product of two vectors is known, along with their magnitudes, we can use the Law of Cosines to find the angle. The law states that for vectors \(\mathbf{a}\) and \(\mathbf{b}\), \(cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|}\), where \(\theta\) is the angle between them. If the dot product is zero, as in the provided exercise, this means that the two vectors are orthogonal and form a \(90^{\circ}\) angle.

To fully understand and visualize this concept, it is advised to draw the vectors on graph paper and construct the angle visually. Another tip is to relate the concept of the angle between vectors to real-life situations, such as using directional signs that cross at right angles, which correspond to a \(90^{\circ}\) intersection. Grasping the relationship between the dot product, vector magnitude, and the angle between vectors is a valuable skill when exploring geometric interpretations of vector arithmetic.