Suggested languages for you:

Americas

Europe

Problem 320

Let $\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots \mathrm{x}_{\mathrm{n}}\right),\left(\mathrm{y}_{1}, \mathrm{y}_{2}, \ldots, \mathrm{y}_{\mathrm{n}}\right),\left(\mathrm{z}_{1}, \mathrm{z}_{2}, \ldots \ldots \mathrm{z}_{\mathrm{n}}\right)$ be three vectors in \(\mathrm{R}^{\mathrm{n}}\). Verify the following properties of the dot product using these vectors : a) \(\mathrm{X} \cdot \mathrm{X} \geq 0 ; \mathrm{X} \cdot \mathrm{X}=0\) if and only if \(\mathrm{X}=0\) b) \(X \cdot Y=Y \cdot X\) c) $(\mathrm{X}+\mathrm{Y}) \cdot \mathrm{Z}=\mathrm{X} \cdot \mathrm{Y}+\mathrm{X} \cdot \mathrm{Z}$ d) $(\mathrm{CX}) \cdot \mathrm{Y}=\mathrm{X} \cdot(\mathrm{cY})=\mathrm{c}(\mathrm{X} \cdot \mathrm{Y})$

Expert verified

In summary, we have verified the properties of the dot product as follows:
a) X.X ≥ 0; X.X = 0 if and only if X = 0: The dot product of a vector with itself is always non-negative, and is 0 if and only if the vector is the zero vector.
b) X ∙ Y = Y ∙ X: The dot product is commutative, meaning it does not matter which vector we perform the operation on first.
c) (X+Y) ∙ Z = X ∙ Z + Y ∙ Z: The dot product is distributive over vector addition, so we can perform the dot product separately on each vector and then add the results.
d) (cX) ∙ Y = X ∙ (cY) = c(X ∙ Y): The dot product remains the same if we multiply one of the vectors by a scalar, and it equals the scalar multipled by the dot product of the original vectors.

What do you think about this solution?

We value your feedback to improve our textbook solutions.

- Access over 3 million high quality textbook solutions
- Access our popular flashcard, quiz, mock-exam and notes features
- Access our smart AI features to upgrade your learning

Chapter 14

Show geometrically that for each angle \(\theta\), the transformation \(\mathrm{T}_{\theta}: \mathrm{R}^{2} \rightarrow \mathrm{R}^{2}\), defined by $(\mathrm{x}, \mathrm{y}) \mathrm{T}_{\theta}=(\mathrm{x} \cos \theta-\mathrm{y} \sin \theta, \mathrm{x} \sin \theta+\mathrm{y} \cos \theta)$, is an orthogonal transformation.

Chapter 14

Compute \(u \cdot v\) where i) \(u=(2,-3,6), \mathrm{v}=(8,2,-3)\); ii) \(\mathrm{u}=(1,-8,0,5), \mathrm{v}=(3,6,4) ;\) iii) \(\mathrm{u}=(3,-5,2,1)\), \(\mathrm{v}=(4,1,-2,5)\)

Chapter 14

Find a vector orthogonal to \(\mathrm{A}=(2,1,-1)\) and \(\mathrm{B}=(1,2,1)\)

Chapter 14

Find the distance between the vectors \(\mathrm{u}\) and \(\mathrm{v}\) where i) \(\mathrm{u}=(1,7), \mathrm{v}=(6,-5) ;\) ii) $\mathrm{u}=(3,-5,4), \mathrm{v}=(6,2,-1)$ iii) \(\mathrm{u}=(5,3,-2,-4,1) \cdot \mathrm{v}=(2,-1,0,-7,2)\)

Chapter 14

Show that the vectors $\mathrm{f}_{1}=(1 / 2, \sqrt{3} / 2), \mathrm{f}_{2}=(\sqrt{3} / 2,-1 / 2)$ form an orthonormal basis for \(E^{2}\). Then find the coordinates of an arbitrary vector \(\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right) \in \mathrm{E}^{2}\).

The first learning app that truly has everything you need to ace your exams in one place.

- Flashcards & Quizzes
- AI Study Assistant
- Smart Note-Taking
- Mock-Exams
- Study Planner