Problem 17 - Algebra and Trigonometry

Question

Determine whether the given vectors are perpendicular. $$ \mathbf{u}=\langle- 2,6\rangle, \quad \mathbf{v}=\langle 4,2\rangle $$

Step-by-Step Solution

Answer

The vectors are not perpendicular.

Step 1: Recall the Condition for Perpendicular Vectors

Two vectors are perpendicular if their dot product is zero. If \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), the dot product is computed as \( u_1 \times v_1 + u_2 \times v_2 \).

Step 2: Compute the Dot Product

Substitute the components of the vectors into the dot product formula. For \( \mathbf{u} = \langle -2, 6 \rangle \) and \( \mathbf{v} = \langle 4, 2 \rangle \), the computation is:\[(-2) \times 4 + 6 \times 2 = -8 + 12 = 4\]

Step 3: Evaluate the Dot Product Result

Since the dot product \( 4 \) is not equal to zero, the vectors \( \mathbf{u} \) and \( \mathbf{v} \) are not perpendicular.

Key Concepts

Dot Product

The dot product is a central concept in vector algebra that allows us to determine the relationship between two vectors. It is a type of vector multiplication. If you have two vectors, \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), their dot product is calculated as:

• \( u_1 \times v_1 + u_2 \times v_2 \)

This operation results in a scalar (a single number), unlike vector multiplication, which results in another vector. The dot product helps us assess whether two vectors are perpendicular. For vectors to be perpendicular, their dot product must equal zero. This property makes it easier to solve a variety of problems in physics, engineering, and computer science.
In the example provided with vectors \( \mathbf{u} = \langle -2, 6 \rangle \) and \( \mathbf{v} = \langle 4, 2 \rangle \), the dot product is calculated to be 4, which means they are not perpendicular.

Vector Components

Understanding vector components is crucial for working with vectors and solving related problems. In the plane, vectors have two components: the horizontal component and the vertical component. For instance, a vector \( \mathbf{u} = \langle u_1, u_2 \rangle \) can be broken down into:

• Horizontal Component: \( u_1 \)
• Vertical Component: \( u_2 \)

These components are the projections of the vector along the x-axis and y-axis, respectively.By representing a vector with its components, you can easily perform arithmetic operations, such as addition, subtraction, and importantly, compute the dot product. When analyzing the given vectors \( \mathbf{u} = \langle -2, 6 \rangle \) and \( \mathbf{v} = \langle 4, 2 \rangle \), you use these components to determine the overall magnitude and direction of the vectors.
This component-wise representation is what makes vector calculations and transformations so powerful and versatile in algebra.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is a unifying thread of almost all mathematics and is used extensively across various domains like physics, engineering, computer graphics, and economics.In the context of vectors, algebra helps us express the vector operations like dot product and vector addition using symbolic representation. By transforming these symbols according to algebraic rules, we can solve equations and inequalities and simplify complex expressions.Here, the algebraic manipulation of vector components allows us to verify the perpendicularity of vectors by calculating their dot product. If the expression simplifies to zero, the vectors are perpendicular. For the vectors \( \mathbf{u} = \langle -2, 6 \rangle \) and \( \mathbf{v} = \langle 4, 2 \rangle \), leveraging algebra shows us the dot product is 4. Therefore, these vectors are not perpendicular. Algebraic principles streamline this process, making it both efficient and straightforward.