Problem 91 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Law of cosines to find the angle \(\alpha\) between the vectors. (Assume \(0^{\circ} \leq \alpha \leq 180^{\circ}\) ). $$\mathbf{v}=\mathbf{i}+\mathbf{j}, \quad \mathbf{w}=2(\mathbf{i}-\mathbf{j})$$

Step-by-Step Solution

Answer

The angle \(\alpha\) between the vectors \(\mathbf{v}\) and \(\mathbf{w}\) is \(90^{\circ}\).

Step 1: Calculate the dot product

The dot product of two vectors \(\mathbf{v}\) and \(\mathbf{w}\) is given by the sum of the product of their respective components. In this case, \(\mathbf{v} \cdot \mathbf{w} = (1)(2) + (1)(-2) = 0.

Step 2: Calculate the magnitude of the vectors

The magnitude (or length) of a vector is given by the square root of the sum of the squares of its components. For \(\mathbf{v}\), this is \(\lVert\mathbf{v}\rVert = \sqrt{(1)^2 + (1)^2} = \sqrt{2}\). For \(\mathbf{w}\), this is \(\lVert\mathbf{w}\rVert = \sqrt{(2)^2 + (-2)^2} = 2\sqrt{2}.

Step 3: Use the Law of cosines to calculate the angle

The Law of cosines can now be used. We have \(\cos(\alpha) = \frac{{\mathbf{v} \cdot \mathbf{w}}}{\lVert\mathbf{v}\rVert \lVert\mathbf{w}\rVert} = \frac{0}{\sqrt{2} \cdot 2\sqrt{2}} = 0\). This gives \(\alpha = \arccos(0) = 90^{\circ}\).

Key Concepts

Understanding Vectors

Vectors are fundamental objects in mathematics and physics. They have both a magnitude and a direction, which makes them very useful for representing quantities like velocity, force, and displacement.
A vector is often written in terms of its components, such as vector \( \mathbf{v} = \mathbf{i} + \mathbf{j} \), where \( \mathbf{i} \) and \( \mathbf{j} \) are the unit vectors along the x and y axes respectively.
Vectors can be visualized as arrows in a coordinate system:

• The length of the arrow represents its magnitude.
• The direction in which the arrow points illustrates the vector's direction.

Understanding vectors and their operations is crucial for further topics like dot products and determining angles between vectors.

The Dot Product

The dot product (also known as the scalar product) is an algebraic operation that takes two equal-length sequences of numbers, usually coordinate vectors, and returns a single number. This product is calculated by multiplying corresponding components and then adding those products.
For example, with vectors \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{w} = c\mathbf{i} + d\mathbf{j} \), their dot product is given by: \[ \mathbf{v} \cdot \mathbf{w} = ac + bd \] The resulting scalar value provides information about the magnitude of the vectors' alignment. If the dot product is:

• Zero, the vectors are perpendicular.
• Positive, they point in similar directions.
• Negative, they point in opposite directions.

The dot product is central to understanding angles between vectors as it relates to the cosine of the angle between the vectors.

Magnitude of a Vector

The magnitude of a vector measures how long it is without considering its direction. It is commonly denoted by \( \lVert \mathbf{v} \rVert \) and calculated as the square root of the sum of the squares of its components.
For a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \), its magnitude is: \[ \lVert \mathbf{v} \rVert = \sqrt{a^2 + b^2} \] This formula derives from the Pythagorean theorem since vector components form a right triangle.
Understanding how to compute the magnitude of vectors is crucial for performing more advanced operations, such as calculating the angle between them using the Law of Cosines.

Angle Between Vectors

The angle between two vectors is an essential concept in vector algebra. It tells us how much two vectors diverge or converge in direction. The Law of Cosines helps us find this angle when we know the magnitudes of the vectors and their dot product.
For vectors \( \mathbf{v} \) and \( \mathbf{w} \), the cosine of the angle \( \alpha \) between them is calculated by the formula: \[ \cos(\alpha) = \frac{\mathbf{v} \cdot \mathbf{w}}{\lVert \mathbf{v} \rVert \lVert \mathbf{w} \rVert} \] Once \( \cos(\alpha) \) is known, the angle \( \alpha \) can be found using the inverse cosine function: \( \alpha = \arccos(\cos(\alpha)) \).
This concept is very useful in physics and engineering, where the relationship between directions is essential for understanding systems.