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Chapter 8: Problem 11

Find the magnitude and direction of the vector $$ \langle 0,4\rangle $$

Short Answer

Expert verified
The magnitude is 4 and the direction is 90掳.

Step by step solution

01

Understand the Problem

We need to find both the magnitude and direction of the vector \( \langle 0,4 \rangle \). A vector in this form indicates its components along the x-axis (0) and y-axis (4).
02

Find the Magnitude

The magnitude of a vector \( \langle a, b \rangle \) is given by the formula \( \sqrt{a^2 + b^2} \). For our vector \( \langle 0, 4 \rangle \), this becomes \( \sqrt{0^2 + 4^2} = \sqrt{16} = 4 \).
03

Determine the Direction

The direction of a vector in the plane is given by the angle \( \theta \) which the vector makes with the positive x-axis. This angle can be found using the arctangent formula \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \). Here, \( a = 0 \) and \( b = 4 \), so \( \theta = \tan^{-1}\left(\frac{4}{0}\right) \). Since division by zero is undefined, and the vector points straight up along the y-axis, \( \theta = 90^\circ \).
04

State the Final Answer

The magnitude of the vector is 4, and its direction is \( 90^\circ \) from the positive x-axis.

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Key Concepts

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

Magnitude of a Vector
The **magnitude of a vector** is a measure of its length or size. It describes how long or large the vector is, essentially giving it a 'distance' from the origin point in a coordinate system. For a vector represented by the coordinates \( \langle a, b \rangle \), the magnitude can be calculated using the formula:
  • \( \sqrt{a^2 + b^2} \)
Think of it like finding the hypotenuse of a right triangle where the vector components \( a \) and \( b \) serve as the two other sides. For instance, if we have the vector \( \langle 0, 4 \rangle \), its magnitude is simply \( \sqrt{0^2 + 4^2} = \sqrt{16} = 4 \). This means the vector has a `length` of 4 units.
Direction of a Vector
The **direction of a vector** gives us insight into which way the vector is pointing. Mathematically, this is expressed as the angle that the vector forms with the positive x-axis. This helps us understand not just how far, but where the vector is pointing in space.
For example, the vector \( \langle 0, 4 \rangle \) points directly upwards on the y-axis.
The angle in this case, since it points directly up, is \( 90^\circ \). Unlike the magnitude, which gives us size, the direction tells us orientation.
Arctangent Formula
The **arctangent formula** is crucial when it comes to finding the direction of a vector. This function helps us calculate the angle \( \theta \) when you have a right triangle formed by the components of a vector. The formula typically looks like this:
  • \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \)
In our scenario with the vector \( \langle 0, 4 \rangle \), the component \( a = 0 \) causes a division by zero situation, which is undefined in regular arithmetic. However, since the vector lies along the y-axis, the angle is intuitively \( \theta = 90^\circ \) because it is pointing straight up.
Vector Components
Vectors are broken down into **components**, which define its behavior in a multidimensional space. When we discuss a vector's components, we're talking about its influence along the x-axis and y-axis, and often further axes in higher dimensions.
In simpler terms, the vector components \( \langle a, b \rangle \) can be imagined as how far and in which direction one must travel along the x and y-axes starting from the origin.For our vector \( \langle 0, 4 \rangle \):
  • The x-component is 0, meaning it does not move along the x-axis at all.
  • The y-component is 4, meaning it moves 4 units straight up the y-axis from the origin.
These components are crucial as they directly inform both the magnitude and direction of the vector.

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