Problem 18 - Algebra and Trigonometry

Question

Determine whether the given vectors are perpendicular. $$ \mathbf{u}=2 \mathbf{i}, \quad \mathbf{v}=-7 \mathbf{j} $$

Step-by-Step Solution

Answer

The vectors are perpendicular because their dot product is zero.

Step 1: Understanding the Concept

Two vectors are perpendicular if and only if their dot product is zero. This is based on the geometric interpretation of the dot product, which is related to the cosine of the angle between the vectors.

Step 2: Identifying the Components of the Vectors

We need to express each vector in terms of its component form. The vector \( \mathbf{u} = 2\mathbf{i}\) is equivalent to \((2,0)\), and the vector \( \mathbf{v} = -7\mathbf{j}\) is equivalent to \((0,-7)\).

Step 3: Calculating the Dot Product

The dot product of two vectors \( \mathbf{u} = (a,b)\) and \( \mathbf{v} = (c,d)\) is given by the formula \(a \cdot c + b \cdot d\). For our vectors, the dot product is \(2 \cdot 0 + 0 \cdot (-7) = 0\).

Step 4: Conclusion on Perpendicularity

Since the dot product equals zero, we can conclude that the vectors \( \mathbf{u} = 2\mathbf{i}\) and \( \mathbf{v} = -7\mathbf{j}\) are perpendicular.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector algebra. It is utilized to determine the relationship between two vectors. The dot product adds up the products of the corresponding components of two vectors. In formulaic terms, for vectors \(\mathbf{a} = (a_1, a_2)\) and \(\mathbf{b} = (b_1, b_2)\), the dot product is calculated as \(a_1 \cdot b_1 + a_2 \cdot b_2\).
This operation doesn't produce a vector but a scalar. The result helps discover various vector properties, such as their angle and orientation. The most intriguing aspect of the dot product is its link to angles. The dot product is zero when two vectors are perpendicular. This happens because the cosine of a 90-degree angle is zero, indicating no projection of one vector onto another.

Vector Components

Vectors are often described via their components. These components break the vector down into simpler elements along the axes of a coordinate system. In the exercise, we had the vectors expressed with the standard unit vectors \(\mathbf{i}\) and \(\mathbf{j}\).
In a two-dimensional space, any vector can be expressed as \((x, y)\), which are its components along the x-axis and y-axis, respectively. For example:

• \(\mathbf{u}=2\mathbf{i} = (2,0)\), indicating no movement along the y-axis and full movement along the x-axis.
• \(\mathbf{v}=-7\mathbf{j} = (0,-7)\), suggesting no movement along the x-axis and full movement along the negative y-axis.

These components assist in calculating operations like addition, subtraction, and most importantly here, the dot product.

Geometric Interpretation of Vectors

Vectors have both magnitude and direction. Geometrically, they are represented as arrows pointing in specific directions with specific lengths. These characteristics can be tied back to fundamental operations, such as the dot product.
The dot product allows us to connect algebraic and geometric interpretations of vectors. When the dot product of two vectors equals zero, it implies that their cosine angle is zero, which geometrically means the vectors are perpendicular. This gives a handy criterion to verify perpendicularity quickly without needing to visually measure angles.
In higher dimensions, the geometric interpretation stays similar. Understanding vectors geometrically helps grasp complex vector operations and interpret their meaningful applications across many fields like physics and engineering.