Suggested languages for you:

Americas

Europe

Problem 316

# 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) $$\mathrm{u}=(3,-5,2,1)$$, $$\mathrm{v}=(4,1,-2,5)$$

Expert verified
i) The dot product of $$u=(2,-3,6)$$ and $$v=(8,2,-3)$$ is -8. ii) The dot product of $$u=(1,-8,0,5)$$ and $$v=(3,6,4)$$ is undefined. iii) The dot product of $$u=(3,-5,2,1)$$ and $$v=(4,1,-2,5)$$ is 8.
See the step by step solution

## Step 1: Write the formula for dot product

The dot product formula for two vectors u and v is: $u \cdot v = u_1v_1 + u_2v_2 + u_3v_3$

## Step 2: Compute the dot product

Using the given vectors and the formula, compute the dot product as follows: $$(2)(8) + (-3)(2) + (6)(-3)$$ Which results in: $$16 - 6 - 18$$

## Step 3: Simplify the expression

Simplify the obtained expression to get the final answer: $$16 - 6 - 18 = -8$$ The dot product of $$u=(2,-3,6)$$ and $$v=(8,2,-3)$$ is -8. **Case ii: Dot product of $$u=(1,-8,0,5)$$ and $$v=(3,6,4)$$** Since the vectors have different numbers of components, their dot product is undefined. **Case iii: Dot product of $$u=(3,-5,2,1)$$ and $$v=(4,1,-2,5)$$**

## Step 1: Write the formula for dot product

The dot product formula for two vectors u and v is: $u \cdot v = u_1v_1 + u_2v_2 + u_3v_3 + u_4v_4$

## Step 2: Compute the dot product

Using the given vectors and the formula, compute the dot product as follows: $$(3)(4) + (-5)(1) + (2)(-2) + (1)(5)$$ Which results in: $$12 - 5 - 4 + 5$$

## Step 3: Simplify the expression

Simplify the obtained expression to get the final answer: $$12 - 5 - 4 + 5 = 8$$ The dot product of $$u=(3,-5,2,1)$$ and $$v=(4,1,-2,5)$$ is 8.

We value your feedback to improve our textbook solutions.

## Access millions of textbook solutions in one place

• Access over 3 million high quality textbook solutions
• Access our popular flashcard, quiz, mock-exam and notes features

## Join over 22 million students in learning with our Vaia App

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