Problem 13
Question
Show that the vectors \(\mathbf{a}=\mathbf{i}-\mathbf{j}, \mathbf{b}=\mathbf{i}+\mathbf{j}\), and \(\mathbf{c}=2 \mathbf{k}\) are mutually orthogonal, that is, each pair of vectors is orthogonal.
Step-by-Step Solution
Verified Answer
The vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are mutually orthogonal as all pairs have a dot product of zero.
1Step 1: Understanding Orthogonal Vectors
Two vectors are orthogonal if their dot product is zero. To show that the vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are mutually orthogonal, we need to calculate the dot product for each pair of vectors and check if the result is zero.
2Step 2: Calculate \(\mathbf{a} \cdot \mathbf{b}\)
The vector \(\mathbf{a} = \mathbf{i} - \mathbf{j}\) and \(\mathbf{b} = \mathbf{i} + \mathbf{j}\). The dot product is calculated as follows:\[\mathbf{a} \cdot \mathbf{b} = (\mathbf{i} - \mathbf{j}) \cdot (\mathbf{i} + \mathbf{j}) = \mathbf{i} \cdot \mathbf{i} + \mathbf{i} \cdot \mathbf{j} - \mathbf{j} \cdot \mathbf{i} - \mathbf{j} \cdot \mathbf{j}\\]Using properties of dot products, \(\mathbf{i} \cdot \mathbf{i} = 1\), \(\mathbf{j} \cdot \mathbf{j} = 1\), \(\mathbf{i} \cdot \mathbf{j} = 0\) and \(\mathbf{j} \cdot \mathbf{i} = 0\):\[1 + 0 - 0 - 1 = 0\]Since \(\mathbf{a} \cdot \mathbf{b} = 0\), \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal.
3Step 3: Calculate \(\mathbf{a} \cdot \mathbf{c}\)
The vector \(\mathbf{c} = 2\mathbf{k}\). The dot product is calculated as follows:\[\mathbf{a} \cdot \mathbf{c} = (\mathbf{i} - \mathbf{j}) \cdot 2\mathbf{k} = 2(\mathbf{i} \cdot \mathbf{k}) - 2(\mathbf{j} \cdot \mathbf{k})\]Using properties of dot products, \(\mathbf{i} \cdot \mathbf{k} = 0\) and \(\mathbf{j} \cdot \mathbf{k} = 0\):\[2(0) - 2(0) = 0\]Since \(\mathbf{a} \cdot \mathbf{c} = 0\), \(\mathbf{a}\) and \(\mathbf{c}\) are orthogonal.
4Step 4: Calculate \(\mathbf{b} \cdot \mathbf{c}\)
The dot product is calculated as follows:\[\mathbf{b} \cdot \mathbf{c} = (\mathbf{i} + \mathbf{j}) \cdot 2\mathbf{k} = 2(\mathbf{i} \cdot \mathbf{k}) + 2(\mathbf{j} \cdot \mathbf{k})\]Using properties of dot products, \(\mathbf{i} \cdot \mathbf{k} = 0\) and \(\mathbf{j} \cdot \mathbf{k} = 0\):\[2(0) + 2(0) = 0\]Since \(\mathbf{b} \cdot \mathbf{c} = 0\), \(\mathbf{b}\) and \(\mathbf{c}\) are orthogonal.
Key Concepts
Dot ProductVector MathematicsMutual Orthogonality
Dot Product
The dot product is a key concept in vector mathematics that helps determine relationships between vectors. It is denoted by \( \cdot \) and involves multiplying corresponding components of two vectors and summing those products.
For example, for vectors \( \mathbf{a} = [a_1, a_2, a_3] \) and \( \mathbf{b} = [b_1, b_2, b_3] \), the dot product is \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).
The dot product results in a scalar value, and importantly, indicates mutual orthogonality when it equals zero.
To check orthogonality, the dot product between the vectors \( \mathbf{i} - \mathbf{j} \) and \( \mathbf{i} + \mathbf{j} \) was calculated, yielding zero, thus confirming they are orthogonal.
Similarly, other pairs \( \mathbf{a} \) and \( \mathbf{c} \), and \( \mathbf{b} \) and \( \mathbf{c} \) also showed a result of zero in their dot products.
For example, for vectors \( \mathbf{a} = [a_1, a_2, a_3] \) and \( \mathbf{b} = [b_1, b_2, b_3] \), the dot product is \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \).
The dot product results in a scalar value, and importantly, indicates mutual orthogonality when it equals zero.
To check orthogonality, the dot product between the vectors \( \mathbf{i} - \mathbf{j} \) and \( \mathbf{i} + \mathbf{j} \) was calculated, yielding zero, thus confirming they are orthogonal.
Similarly, other pairs \( \mathbf{a} \) and \( \mathbf{c} \), and \( \mathbf{b} \) and \( \mathbf{c} \) also showed a result of zero in their dot products.
Vector Mathematics
Vector mathematics is a field that deals with quantities having both direction and magnitude. Vectors are often considered in 2D or 3D space and have components along the coordinate axes.
For instance, a vector \( \mathbf{a} \) can be expressed as \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \), where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are unit vectors in the \( x, y, z \) directions, respectively.
Vectors can be added, subtracted, and multiplied by scalars to derive new vectors, and the dot product allows us to gain insights into their spatial relationships.
Understanding how operations like dot product and cross product work helps us solve real-world problems, such as determining forces in physics or rendering graphics in computer science.
This problem demonstrates how simple vector operations are applied to determine orthogonality, emphasizing the practical utility of these mathematical tools.
For instance, a vector \( \mathbf{a} \) can be expressed as \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \), where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are unit vectors in the \( x, y, z \) directions, respectively.
Vectors can be added, subtracted, and multiplied by scalars to derive new vectors, and the dot product allows us to gain insights into their spatial relationships.
Understanding how operations like dot product and cross product work helps us solve real-world problems, such as determining forces in physics or rendering graphics in computer science.
This problem demonstrates how simple vector operations are applied to determine orthogonality, emphasizing the practical utility of these mathematical tools.
Mutual Orthogonality
Mutual orthogonality occurs when each pair from a set of vectors is orthogonal.
Orthogonality means the dot product of any two vectors equals zero, which geometrically signifies that the vectors are perpendicular to each other.
In the problem, the vectors \( \mathbf{a} = \mathbf{i} - \mathbf{j} \), \( \mathbf{b} = \mathbf{i} + \mathbf{j} \), and \( \mathbf{c} = 2 \mathbf{k} \), are mutually orthogonal.
This means every possible pair taken from these vectors results in a dot product of zero:
By analyzing mutual orthogonality, one can confirm that vectors behave independently in different dimensions, a concept frequently utilized in physics and engineering.
Orthogonality means the dot product of any two vectors equals zero, which geometrically signifies that the vectors are perpendicular to each other.
In the problem, the vectors \( \mathbf{a} = \mathbf{i} - \mathbf{j} \), \( \mathbf{b} = \mathbf{i} + \mathbf{j} \), and \( \mathbf{c} = 2 \mathbf{k} \), are mutually orthogonal.
This means every possible pair taken from these vectors results in a dot product of zero:
- \( \mathbf{a} \cdot \mathbf{b} = 0 \)
- \( \mathbf{a} \cdot \mathbf{c} = 0 \)
- \( \mathbf{b} \cdot \mathbf{c} = 0 \)
By analyzing mutual orthogonality, one can confirm that vectors behave independently in different dimensions, a concept frequently utilized in physics and engineering.
Other exercises in this chapter
Problem 13
For the three-dimensional vectors \(\mathbf{u}\) and \(\mathbf{v}\) in Problems 13-16, find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\math
View solution Problem 13
Find \(D_{t} \mathbf{r}(t)\) and \(D_{t}^{2} \mathbf{r}(t)\) for each of the following: (a) \(\mathbf{r}(t)=(3 t+4)^{3} \mathbf{i}+e^{t^{2}} \mathbf{j}+\mathbf{
View solution Problem 13
Name and sketch the graph of each of the following equations in three-space. $$ x^{2}-z^{2}+y=0 $$
View solution Problem 13
In Problems 13-16, complete the squares to find the center and \(\mathrm{ra}\) dius of the sphere whose equation is given (see Example 2). x^{2}+y^{2}+z^{2}-12
View solution