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Chapter 14: Chapter 14

Problem 316

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

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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.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

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.

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