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Chapter 3: Problem 51

The sum of two vectors, \(\vec{A}+\vec{B},\) is perpendicular to their difference, \(\vec{A}-\vec{B}\). How do the vectors' magnitudes compare?

Short Answer

Expert verified
The magnitudes of vector \(\vec{A} \) and vector \(\vec{B} \) are equal.

Step by step solution

01

Understanding the property of orthogonal vectors

Vectors \(\vec{A}+\vec{B}\) and \(\vec{A}-\vec{B}\) are orthogonal if their dot product is 0. Thus, we can write that: (\vec{A}+\vec{B})\cdot(\vec{A}-\vec{B}) = 0
02

Expand the Dot Product using Distributive Laws

We can expand the left side of the equation using the distributive property of dot product, which says that \(\vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{A} = 2 \vec{A} \cdot \vec{B} \). So: (\vec{A}+\vec{B})\cdot(\vec{A}-\vec{B}) = \vec{A}\cdot\vec{A} - \vec{A}\cdot\vec{B} + \vec{B}\cdot\vec{A} - \vec{B}\cdot\vec{B} = \vec{A}\cdot\vec{A} - 2 \vec{A}\cdot\vec{B} - \vec{B}\cdot\vec{B} = 0.
03

Change in terms of magnitudes

Next, we will convert this to the language of magnitudes. Recall that the dot product of a vector with itself is simply the square of its magnitude. Thus we have: |\vec{A}|^2 - 2 \vec{A}\cdot\vec{B} - |\vec{B}|^2 = 0.
04

Express the Dot Product in terms of magnitudes and angles

In order to create a formula wherein we can eliminate the dot product between the vectors \(\vec{A} \) and \(\vec{B} \), we express this term using the relation \(\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta ) \), where \(\theta\) is the angle between the vectors. We substitute this into the equation: |\vec{A}|^2 - 2|\vec{A}||\vec{B}|\cos(\theta ) - |\vec{B}|^2 = 0.
05

Solve for the condition on magnitudes given by the problem

Given the vectors \(\vec{A} \) and \(\vec{B} \) are perpendicular with their sum, the angle between the vectors is 90 degrees and cosine of 90 degrees is 0. Therefore, the equation simplifies to : |\vec{A}|^2 - |\vec{B}|^2 = 0,\ suggesting that |\vec{A}| = |\vec{B}|

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Most popular questions from this chapter

An object is initially moving in the \(x\) -direction at \(4.5 \mathrm{m} / \mathrm{s}\), when it undergoes an acceleration in the \(y\) -direction for a period of \(18 \mathrm{s}\) If the object moves equal distances in the \(x\) - and \(y\) -directions during this time, what's the magnitude of its acceleration?

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