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Chapter 14: Problem 3

Two vectors have moduli 7 and 13 respectively. The angle between them is \(45^{\circ}\). Evaluate their scalar product.

Short Answer

Expert verified
Answer: The scalar product of the two given vectors is \(A \cdot B = 91 \cdot \frac{\sqrt{2}}{2}\).

Step by step solution

01

Convert degrees to radians

Since trigonometric functions often deal with radians, convert the angle from degrees to radians: \(\theta = 45^{\circ} \cdot \frac{\pi}{180} = \frac{\pi}{4}\)
02

Write down the magnitudes and angle

Now, we have the magnitudes of both vectors and the angle between them: |A| = 7, |B| = 13 and \(\theta = \frac{\pi}{4}\)
03

Use the scalar product formula

Using the scalar product formula, we can calculate the scalar product of the two vectors: \(A \cdot B = |A||B|\cos\theta\)
04

Insert the given values into the formula

Put the given values of magnitudes and angle into the formula: \(A \cdot B = 7 \cdot 13 \cdot \cos\frac{\pi}{4}\)
05

Compute the scalar product

Now, calculate the scalar product using the given values: \(A \cdot B = 7 \cdot 13 \cdot \frac{\sqrt{2}}{2} = 91 \cdot \frac{\sqrt{2}}{2}\) So, the scalar product of the two given vectors is: \(A \cdot B = 91 \cdot \frac{\sqrt{2}}{2}\).

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