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Chapter 1: Problem 10

If \(\vec{c}\) is a unit voctor and cqual to the sum of \(\vec{a}\) and \(\vec{b}\), the magnitude of differenec belwecn \(\vec{a}\) and \(\vec{b}\) is (a) 1 (b) \(\sqrt{2}\) (c) \(\sqrt{3}\) (d) \(\frac{1}{\sqrt{3}}\)

Short Answer

Expert verified
The magnitude of the difference is \(\sqrt{2}\). Option (b) is correct.

Step by step solution

01

Understanding Unit Vector Definition

A unit vector is a vector with a magnitude of 1. Mathematically, if \( \vec{c} \) is a unit vector, then \( |\vec{c}| = 1 \).
02

Express \( \vec{c} \) in terms of \( \vec{a} \) and \( \vec{b} \)

Given \( \vec{c} = \vec{a} + \vec{b} \). Since \( \vec{c} \) is a unit vector: \( |\vec{c}| = |\vec{a} + \vec{b}| = 1 \).
03

Square the Magnitude Equation

Squaring the equation from Step 2, we get: \[ (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = 1^2 \] This simplifies to \[ \vec{a} \cdot \vec{a} + 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = 1 \].
04

Set Terms for Difference in Magnitude

We want \( |\vec{a} - \vec{b}| \). Use the identity \[ |\vec{a} - \vec{b}|^2 = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b}) \]. This equals \[ \vec{a} \cdot \vec{a} - 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} \].
05

Solve for \( |\vec{a} - \vec{b}| \)

Substitute the values from the equation in Step 3. We already have \( \vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{b} = 1 \) and \( \vec{a} \cdot \vec{b} = \frac{-1}{2} \). Now substitute: \[ |\vec{a} - \vec{b}|^2 = (1) + 2 \times \frac{1}{2} = 2 \]Therefore, \( |\vec{a} - \vec{b}| = \sqrt{2} \).
06

Conclusion

The magnitude of the difference between vectors \( \vec{a} \) and \( \vec{b} \) is \( \sqrt{2} \). Therefore, the correct option is (b).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Vector
A unit vector is a fundamental concept in vector analysis. It is a vector that has a magnitude of 1, essentially representing a direction without any length. This property makes unit vectors incredibly useful for defining directions. For example, when we say a vector represents 10 meters but in an unspecified direction, using a unit vector can clarify this. The unit vector \( \vec{c} \) ensures that regardless of the dimensions, the direction remains clear.
  • Magnitude is always 1: \( |\vec{c}| = 1 \).
  • Often used to express other vectors in terms of direction.
  • Commonly denoted with a hat, e.g., \( \hat{i} \) for the \( x\)-axis unit vector.
Understanding this concept will cushion your learning and allow you to transition smoothly to more complex vector operations, such as combining vectors.
Vector Addition
Vector addition combines two or more vectors to create a new vector. This operation is vital to comprehending how vectors interact and relate in different spaces. When adding vectors such as \( \vec{a} \) and \( \vec{b} \), as in the given problem where \( \vec{c} = \vec{a} + \vec{b} \), the result is called the resultant vector.
  • Vectors are added component-wise: \( \vec{a} + \vec{b} = (a_1+b_1, a_2+b_2,...) \).
  • The resultant vector shares a geometric space and direction influenced by the original vectors.
  • The magnitude and direction of the resultant vector depend on the relative directions and magnitudes of the original vectors.
Visualizing this helps to comprehend more complex operations, like subtracting vectors or finding differences.
Magnitude of a Vector
The magnitude of a vector, denoted as \( |\vec{a}| \), informs us of the "length" or size of the vector. This is a scalar quantity 鈥� it has no direction, only size. Calculating magnitude is essential for understanding the scale and impact of a vector in its space. It frequently comes into play in physics, engineering, and computer graphics.
  • Calculated using the Pythagorean theorem: \( |\vec{a}| = \sqrt{a_1^2 + a_2^2 + ...} \).
  • Provides information about the scale relative to other vectors.
  • Informs the weight or impact when added or subtracted from other vectors.
Magnitude is a stepping stone to further vector operations, ensuring accuracy and thorough understanding of vector behavior.
Dot Product of Vectors
The dot product, or scalar product, is an operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation tells us the extent to which two vectors point in the same or opposite directions, providing insight into their geometric relationship.
  • Formula: \( \vec{a} \cdot \vec{b} = a_1 \cdot b_1 + a_2 \cdot b_2 + ... \).
  • If the dot product is zero, the vectors are orthogonal (perpendicular).
  • Helps in evaluating projections and angles between vectors.
In the provided exercise, it's applied to find the magnitude difference, highlighting how dot product significantly affects and defines vector scopes.

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Most popular questions from this chapter

Vector \(\vec{P}-6 \hat{i}+4 \sqrt{2} \hat{j}+4 \sqrt{2} \hat{k}\) makes angle from \(z\) -axis equal to (a) \(\cos ^{-1}\left(\frac{\sqrt{2}}{5}\right)\) (b) \(\cos ^{-1}(2 \sqrt{2})\) (c) \(\cos ^{-1}\left(\frac{2 \sqrt{2}}{5}\right)\) (d) None

Find the value of \(\lambda\) so that the two vectors \(2 \hat{i}+3 \hat{j}-\hat{k}\) and \(-4 \hat{i}-6 \hat{j}+\lambda \hat{k}\) are: (i) parallel; (ii) perpendicular to cach other.

If \(\vec{a}\) and \(\vec{b}\) are two unit vectors such that \(\vec{a}+2 \vec{b}\) and \(5 \vec{a}-4 \vec{b}\) are perpendicular to each other, then the angle between \(\vec{a}\) and \(\vec{b}\) is (a) \(45^{\circ}\) (b) \(60^{\circ}\) (c) \(\cos ^{-1}\left(\frac{1}{3}\right)\) (d) \(\cos ^{-1}\left(\frac{2}{7}\right)\)

Show that \((\vec{A}+\vec{B}) \times(\vec{A}-\vec{B})-2(\vec{B} \times \vec{A})\)

Statement- \(\mathbf{1}:\) 'lhe value of \(\hat{i} \cdot(\hat{j} \times \hat{k})|\hat{j} \cdot(\hat{k} \times \hat{i})| \dot{k} \cdot(\hat{i} \times \hat{j})\) is equal to 3 Statement- \(2:\) If \(\hat{a}, \hat{b}, \hat{c}\) are mutually perpendicular unit vectors, then \(\lfloor\hat{a}, \hat{b}, \hat{c}\rfloor-1\)
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