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Chapter 7: Problem 27

Find the magnitude of each of the following vectors. $$\mathbf{W}=\mathbf{i}+2 \mathbf{j}$$

Short Answer

Expert verified
The magnitude of vector \( \mathbf{W} \) is \( \sqrt{5} \).

Step by step solution

01

Understand Vector Components

The vector \( \mathbf{W} = \mathbf{i} + 2\mathbf{j} \) is given in component form, where \( \mathbf{i} \) is the unit vector along the x-axis and \( \mathbf{j} \) is the unit vector along the y-axis. In this form, \( \mathbf{W} = (1, 2) \).
02

Apply the Magnitude Formula

To find the magnitude of a vector \( \mathbf{a} = (a_1, a_2) \), use the formula: \[\| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2}\] For \( \mathbf{W} = (1, 2) \), this becomes:\[\| \mathbf{W} \| = \sqrt{1^2 + 2^2}\]
03

Calculate the Squares

Calculate \( 1^2 \) and \( 2^2 \): - \( 1^2 = 1 \)- \( 2^2 = 4 \)This gives us a sum of: \[ 1 + 4 = 5 \]
04

Compute the Square Root

Now find the square root of the sum:\[ \sqrt{5} \]The magnitude of vector \( \mathbf{W} \) is \( \sqrt{5} \).

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Key Concepts

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

Vector Components
Vectors are mathematical entities used to represent quantities that have both magnitude (size) and direction. A vector is often written in terms of its components along the x-axis and y-axis. These components indicate how far the vector stretches along each axis. For instance, the vector \( \mathbf{W} = \mathbf{i} + 2\mathbf{j} \) consists of:
  • \( \mathbf{i} \): the unit vector in the x-direction, indicating the vector moves one unit to the right.
  • \( 2\mathbf{j} \): two times the unit vector in the y-direction, indicating the vector moves two units upwards.
These components can be bundled into an ordered pair, written as \( (1, 2) \). This notation simplifies understanding and calculations. It鈥檚 essentially a map showing exactly where the head of the vector lands from its starting point.
Magnitude Formula
The magnitude of a vector is a measure of its length, regardless of direction. For any vector \( \mathbf{a} = (a_1, a_2) \), the magnitude formula is expressed as:\[\| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2}\]This formula derives from the Pythagorean Theorem and provides a straightforward method to find the length of a vector. In the case of vector \( \mathbf{W} = (1, 2) \):
  • Substitute \( a_1 = 1 \) and \( a_2 = 2 \) into the magnitude formula.
  • Calculate \( \sqrt{1^2 + 2^2} \), which simplifies to \( \sqrt{5} \).
The result, \( \sqrt{5} \), reveals the magnitude of the vector, showing how far it stretches from its origin.
Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in mathematics that applies to right-angled triangles. It states:\[c^2 = a^2 + b^2\]Where \( c \) is the hypotenuse, and \( a \) and \( b \) are the triangle's other two sides.In vector mathematics, this concept helps calculate a vector's magnitude. Consider a vector's components as the sides of a right triangle, with the vector itself as the hypotenuse. For \( \mathbf{W} = (1, 2) \):
  • The horizontal component \( 1 \) and the vertical component \( 2 \) act as the triangle's two legs.
  • The magnitude, or hypotenuse, becomes \( \sqrt{1^2 + 2^2} \), which is \( \sqrt{5} \).
Using the Pythagorean Theorem thus elegantly demonstrates the relationship between vector components and their magnitude.

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Most popular questions from this chapter

Use the law of cosines to show that, if \(a^{2}=b^{2}+c^{2}\), then \(A=90^{\circ}\).

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