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Chapter 5: Problem 36

Find a unit vector orthogonal to both u and v. $$\begin{aligned} &\mathbf{u}=(2,-1,3)\\\ &\mathbf{v}=(1,0,-2) \end{aligned}$$

Short Answer

Expert verified
The unit vector orthogonal to both vector \( \mathbf{u} \) and \( \mathbf{v} \) is \( \mathbf{u_{w}} \)

Step by step solution

01

Calculate Cross Product

Calculate the cross product of the two vectors with the formula \( \mathbf{w} = \mathbf{u} \times \mathbf{v} \), thereby using the coordinates from vector \( \mathbf{u} = (2,-1,3) \) and vector \( \mathbf{v} = (1,0,-2) \)
02

Normalization factor

Calculate the normalization factor (magnitude of the cross product vector), denoted N, with the formula \( N = \sqrt{w_{1}^2 + w_{2}^2 + w_{3}^2}\), where \(w_{1}, w_{2}, w_{3}\) are the components of the cross product vector.
03

Compute the Unit Vector

Divide the cross product vector \( \mathbf{w} \) by the normalization factor N, obtaining the unit vector \( \mathbf{u_{w}} = \frac{1}{N} \mathbf{w} \)

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Key Concepts

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

Cross Product
The cross product is a vector operation used in three-dimensional space, often denoted by the symbol \( \times \). It involves two vectors and results in a third vector that is orthogonal, or perpendicular, to both of the original vectors. For vectors \( \mathbf{u} = (2, -1, 3) \) and \( \mathbf{v} = (1, 0, -2) \), the cross product \( \mathbf{w} = \mathbf{u} \times \mathbf{v} \) can be calculated.
  • To find the cross product, set up a 3x3 matrix with the top row as the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), the second row as the components of \( \mathbf{u} \), and the third row as the components of \( \mathbf{v} \).
  • The cross product formula involves applying the determinant of this matrix, which results in \( \mathbf{w} = ((-1)(-2) - (3)(0), (3)(1) - (2)(-2), (2)(0) - (-1)(1)) \).
  • Simplifying this gives us the vector \( \mathbf{w} = (2, 7, 1) \).
This result, \( \mathbf{w} \), is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \). Understanding the cross product is essential in physics and engineering to find torques and rotational forces.
Orthogonal Vectors
Orthogonal vectors are two vectors that meet at a right angle. They are characterized by having a dot product of zero. When working with vectors like \( \mathbf{u} \) and \( \mathbf{v} \), finding a vector orthogonal to both involves using the cross product.
  • Two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal if \( \mathbf{a} \cdot \mathbf{b} = 0 \).
  • The cross product vector, \( \mathbf{w} = \mathbf{u} \times \mathbf{v} \), is naturally orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \).
  • This property makes it useful in constructing perpendicular dimensions or directions, like in 3D modeling and geometry.
Recognizing and using orthogonality can simplify many complex problems in areas such as vector projections and independent bases in linear algebra.
Normalization
Normalization refers to the process of converting a vector into a unit vector, which means it has a magnitude of 1.
  • The magnitude of a vector \( \mathbf{w} = (w_{1}, w_{2}, w_{3}) \) is calculated using the formula: \( N = \sqrt{w_{1}^2 + w_{2}^2 + w_{3}^2} \).
  • Once you have the magnitude, you divide each component of the vector \( \mathbf{w} \) by this magnitude \( N \) to get the unit vector \( \mathbf{u_{w}} = \frac{1}{N} \mathbf{w} \).
  • For our cross product vector \( \mathbf{w} = (2, 7, 1) \), calculating the magnitude gives \( N = \sqrt{2^2 + 7^2 + 1^2} = \sqrt{54} \).
  • Then, normalize \( \mathbf{w} \) to get the unit vector: \( \left( \frac{2}{\sqrt{54}}, \frac{7}{\sqrt{54}}, \frac{1}{\sqrt{54}} \right) \).
This process gives us a direction vector with a magnitude of 1, which is crucial for ensuring consistency when scaling for diverse applications like graphics and physics.
Linear Algebra
Linear Algebra is the branch of mathematics that deals with vectors, vector spaces, and linear transformations. It provides several tools to work with multi-dimensional datasets, which is essential in today's data-driven world.
  • It covers fundamental operations with vectors and matricies, crucial for scientific computations.
  • Understanding concepts like vector spaces, eigenvectors, and eigenvalues can help solve complex systems of equations.
  • Operations such as the cross product allow us to determine perpendicular vectors, a cornerstone in 3D space calculations.
  • Linear algebra concepts are widely applicable, from computer graphics, computational physics to machine learning and more.
By mastering linear algebra, students can develop strong problem-solving skills, which are applicable in both academic and professional settings, facilitating innovation and efficiency.

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Most popular questions from this chapter

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. Prove that\(\|\mathbf{u}+\mathbf{v}\|^{2}+\|\mathbf{u}-\mathbf{v}\|^{2}=2\|\mathbf{u}\|^{2}+2\|\mathbf{v}\|^{2}\)for any vectors \(\mathbf{u}\) and \(\mathbf{v}\) in an inner product space \(V\)

Find the area of the triangle with the given vertices. Use the fact that the area \(A\) of the triangle having \(u\) and \(v\) as adjacent sides is \(A=\frac{1}{2}\|\mathbf{u} \times \mathbf{v}\|\). $$(3,5,7),(5,5,0),(-4,0,4)$$

Prove that \(\mathbf{u} \times \mathbf{v}\) is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\).

Prove Property 2 of Theorem 5.7: If \(\mathbf{u}, \mathbf{v}\) and \(\mathbf{w}\) are vectors in an inner product space \(V,\) then \(\langle\mathbf{u}+\mathbf{v}, \mathbf{w}\rangle=\langle\mathbf{u}, \mathbf{w}\rangle+\langle\mathbf{v}, \mathbf{w}\rangle\)

Find the Fourier approximation with the specified order of the function on the interval \([0,2 \pi]\). \(f(x)=(x-\pi)^{2}, \quad\) third order
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