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Chapter 3: Problem 27

\(\mathbf{u}=2 \mathbf{i}, \quad \mathbf{v}=\mathbf{j}\)

Short Answer

Expert verified
The dot product of the vectors \(\mathbf{u}\) and \(\mathbf{v}\) is 0.

Step by step solution

01

Identify the vectors

The vectors given in the problem are \(\mathbf{u}=2 \mathbf{i}\) and \(\mathbf{v}=\mathbf{j}\). The first vector \(\mathbf{u}\) has components 2 in the x-direction and 0 in the y-direction. The second vector \(\mathbf{v}\) has components 0 in the x-direction and 1 in the y-direction.
02

Compute dot product

Now, compute the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). The dot product is given by the sum of the products of their corresponding components. So, it's \((2 \times 0) + (0 \times 1)\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Components
In the realm of vectors, understanding vector components is crucial. Vector components break down a vector into its respective parts along different axes. For example, when we look at a 2-D vector such as \(\mathbf{u} = 2 \mathbf{i}\) and \(\mathbf{v} = \mathbf{j}\), we describe the vector components with respect to the x-axis and y-axis.
  • The vector \(\mathbf{u} = 2 \mathbf{i}\) has components 2 along the x-axis, denoted by i, and 0 along the y-axis, meaning it does not point in the j direction at all.
  • Similarly, \(\mathbf{v} = \mathbf{j}\) has components 0 along the x-axis (so there's no i component) and 1 along the y-axis.

These components are essential as they allow for precise calculations, such as the dot product, by focusing on the individual contributions of each direction. Understanding each component clearly simplifies the process of vector operations, allowing you to easily visualize and calculate their interactions.
Mathematical Vectors
Mathematical vectors are essential tools for representing quantities that have both magnitudes and directions. They are often depicted with arrows in geometrical space, where the length of the arrow signifies the magnitude and the arrowhead indicates the direction.
  • Vectors can exist in one or more dimensions. In 2-D space, they can have components along the x and y directions.
  • Common notation includes \(\mathbf{i}\) and \(\mathbf{j}\) for representing unit vectors in the x and y directions respectively.

Importantly, vectors can be added, subtracted, or multiplied to find various results, such as force or movement. For instance, in the example \(\mathbf{u} = 2 \mathbf{i}\), you have a vector purely in the x direction. Such vectors are foundational in physics and engineering, simplifying complex calculations and providing insights into different physical scenarios.
Dot Product Calculation
The dot product, also known as the scalar product, is a fundamental operation for mathematical vectors. It provides a scalar result from two vectors by multiplying their corresponding components and summing them up. This calculation helps to determine the degree to which the vectors align with each other.
For vectors \(\mathbf{u} = 2 \mathbf{i}\) and \(\mathbf{v} = \mathbf{j}\), the process is as follows:
  • Multiply the x-components: \(2 \times 0 = 0\)
  • Multiply the y-components: \(0 \times 1 = 0\)
  • Add these products: \(0 + 0 = 0\)

Hence, the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is 0, indicating that the vectors are perpendicular to each other. Remember, a dot product of zero always signals perpendicularity, as there is zero projection of one vector on the other.

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Most popular questions from this chapter

In Exercises 59-62, find two vectors in opposite directions that are orthogonal to the vector u. (There are many correct answers.) $$ \mathbf{u}=\langle-8,3\rangle $$

\(\mathbf{u}=\langle-5,3\rangle, \quad \mathbf{v}=\langle 0,0\rangle\)

\(\|\mathrm{v}\|=10\) \(\mathbf{u}=\langle-10,0\rangle\)

\(\|\mathbf{v}\|=1 \quad \theta=45^{\circ}\)

A sport utility vehicle with a gross weight of 5400 pounds is parked on a slope of \(10^{\circ}\). Assume that the only force to overcome is the force of gravity. Find the force required to keep the vehicle from rolling down the hill. Find the force perpendicular to the hill.
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