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Chapter 11: Problem 11

Show that the vectors \langle 6,3\rangle and \langle-1,2\rangle are orthogonal.

Short Answer

Expert verified
The vectors are orthogonal because their dot product is zero.

Step by step solution

01

Understand the Definition of Orthogonal Vectors

Two vectors are orthogonal if their dot product is zero. The dot product of two vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is calculated as \( a \cdot c + b \cdot d \).
02

Identify the Components of the Given Vectors

The first vector is \( \langle 6, 3 \rangle \), which has components 6 (for \( a \)) and 3 (for \( b \)). The second vector is \( \langle -1, 2 \rangle \), which has components -1 (for \( c \)) and 2 (for \( d \)).
03

Calculate the Dot Product

Using the dot product formula from Step 1, calculate the dot product: \( 6 \cdot (-1) + 3 \cdot 2 = -6 + 6 \).
04

Determine if the Vectors are Orthogonal

Substitute the values obtained in Step 3: \( -6 + 6 = 0 \). Since the dot product is zero, the vectors \( \langle 6, 3 \rangle \) and \( \langle -1, 2 \rangle \) are orthogonal.

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Key Concepts

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

Dot Product
The dot product is a fundamental operation in vector analysis, used to find the relationship between two vectors. To calculate it, multiply the corresponding components of two vectors and then sum those products. It's often represented as \( \vec{a} \cdot \vec{b} \). For example, given vectors \( \langle 6, 3 \rangle \) and \( \langle -1, 2 \rangle \), their dot product is calculated as:
  • Multiply corresponding components: \( 6 \cdot (-1) = -6 \) and \( 3 \cdot 2 = 6 \).
  • Sum these values: \(-6 + 6 = 0\).
A result of zero indicates that the vectors are orthogonal, meaning they are perpendicular to each other in space.
Vector Components
Understanding vector components is essential for calculating the dot product and other vector operations. A vector is defined by its components, typically represented in the form \( \langle a, b \rangle \) in two dimensions, where \( a \) and \( b \) are the individual components along the x and y axes.
For our vectors, \( \langle 6, 3 \rangle \) has components 6 and 3. This means:
  • 6 is the component along the x-axis.
  • 3 is the component along the y-axis.
Similarly, \( \langle -1, 2 \rangle \) has components -1 and 2. Recognizing these components
is crucial for tasks like adding vectors, calculating the dot product, or determining orthogonal relationships.
Vector Analysis
Vector analysis involves multiple techniques to understand and manipulate vectors. It allows us to explore properties such as magnitude, direction, and relationships with other vectors
(like parallelism or orthogonality). In our scenario, analyzing vectors \( \langle 6, 3 \rangle \) and \( \langle -1, 2 \rangle \) through their dot product provided insight into their orthogonality.

To determine if vectors are orthogonal, calculate their dot product. If the result is zero, the vectors are perpendicular:
  • This means no projection of one vector onto the other.
  • They form a right angle.
Overall, vector analysis is a powerful tool used not only in math but also in physics and engineering to solve complex spatial problems.

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Most popular questions from this chapter

Find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1}\). $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j} ; t_{1}=1 $$

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