Problem 110
Question
Determine whether the statement is true or false. Justify your answer.If \(\mathbf{u}=a \mathbf{i}+b \mathbf{j}\) is a unit vector, then \(a^{2}+b^{2}=1\).
Step-by-Step Solution
Verified Answer
The statement is true. If \(\mathbf{u}=a \mathbf{i}+b \mathbf{j}\) is indeed a unit vector, then \(a^{2}+b^{2}\) must equal 1, according to the mathematical definition and properties of a unit vector.
1Step 1: Understand the definition of a unit vector
A unit vector is defined as a vector of magnitude 1. That is, if you consider the vector's length or size, it equates to 1.
2Step 2: Recall the method for calculating the magnitude of a vector
The magnitude or length of a vector \(\mathbf{u} = a\mathbf{i} + b\mathbf{j}\) in two dimensions (2D) is calculated as \(\|\mathbf{u}\| = \sqrt{a^2 + b^2}\).
3Step 3: Check the condition for the vector to be a unit vector
A unit vector must have a magnitude of 1. Substituting the equation from step 2, for the vector to be a unit vector, we must have \(\sqrt{a^2 + b^2} = 1\). Squaring both sides of this equation (to get rid of the square root), we indeed get \(a^2 + b^2 = 1\).
Key Concepts
Vector MagnitudeTwo-Dimensional VectorsPythagorean Theorem
Vector Magnitude
Understanding the magnitude of a vector is crucial when dealing with vector operations in various branches of mathematics and physics. The magnitude of a vector, often symbolized by \( \|\mathbf{u}\| \), represents the 'length' or 'size' of the vector from its initial point to its terminal point, regardless of its direction.
In the context of unit vectors, magnitude is especially important because a unit vector is defined by its magnitude being exactly 1. This is true for vectors in two-dimensional (2D) space, three-dimensional (3D) space, and beyond. The process to calculate magnitude involves using the coordinates of the vector. For a 2D vector \(\textbf{u} = a\textbf{i} + b\textbf{j}\), the magnitude is computed by the square root of the sum of the squared components: \(\|\mathbf{u}\| = \sqrt{a^2 + b^2}\). It's clear to see that mastering this concept is fundamental for confirming whether a given vector can be classified as a unit vector or not.
In the context of unit vectors, magnitude is especially important because a unit vector is defined by its magnitude being exactly 1. This is true for vectors in two-dimensional (2D) space, three-dimensional (3D) space, and beyond. The process to calculate magnitude involves using the coordinates of the vector. For a 2D vector \(\textbf{u} = a\textbf{i} + b\textbf{j}\), the magnitude is computed by the square root of the sum of the squared components: \(\|\mathbf{u}\| = \sqrt{a^2 + b^2}\). It's clear to see that mastering this concept is fundamental for confirming whether a given vector can be classified as a unit vector or not.
Two-Dimensional Vectors
Two-dimensional vectors are mathematical objects defined in a plane by two components, often corresponding to the horizontal (x-axis) and vertical (y-axis) dimensions. They can be represented both graphically, as arrows originating from a point and extending to another, and algebraically, in the form \( \textbf{u} = a\textbf{i} + b\textbf{j} \) where \(a\) and \(b\) are the vector's components along the x and y axes respectively, and \(\textbf{i}\) and \(\textbf{j}\) are the unit vectors in the horizontal and vertical directions.
When dealing with 2D vectors, it is important to understand how to perform operations such as addition, subtraction, and especially, how to calculate their magnitude. The ability to calculate magnitude feeds directly into other operations, like scaling a vector, where maintaining the direction while changing the length is needed. Unit vectors are essentially scaled to have a magnitude of 1, making them a cornerstone in vector arithmetic.
When dealing with 2D vectors, it is important to understand how to perform operations such as addition, subtraction, and especially, how to calculate their magnitude. The ability to calculate magnitude feeds directly into other operations, like scaling a vector, where maintaining the direction while changing the length is needed. Unit vectors are essentially scaled to have a magnitude of 1, making them a cornerstone in vector arithmetic.
Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in geometry, connecting the lengths of the sides of a right-angled triangle. According to the theorem, if you have a right triangle with sides of length \(a\) and \(b\), and the hypotenuse (the side opposite the right angle) having length \(c\), then \(a^2 + b^2 = c^2\). This relation is extremely useful in a wide variety of mathematical contexts, including when working with two-dimensional vectors.
In the context of vectors, the components \(a\) and \(b\) can be regarded as the sides of a right triangle, while the hypotenuse represents the vector's magnitude. For unit vectors, since the magnitude is 1, the Pythagorean theorem gives us the neat condition that \(a^2 + b^2 = 1\), embodying the idea that the sum of the squared components of the unit vector in 2D must equal 1. Grasping this theorem not only aids in understanding unit vectors but also builds a bridge to more complex geometric problems.
In the context of vectors, the components \(a\) and \(b\) can be regarded as the sides of a right triangle, while the hypotenuse represents the vector's magnitude. For unit vectors, since the magnitude is 1, the Pythagorean theorem gives us the neat condition that \(a^2 + b^2 = 1\), embodying the idea that the sum of the squared components of the unit vector in 2D must equal 1. Grasping this theorem not only aids in understanding unit vectors but also builds a bridge to more complex geometric problems.
Other exercises in this chapter
Problem 109
Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(-1+i)^{6}$$
View solution Problem 110
Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(3-3 i)^{8}$$
View solution Problem 111
Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$2(\sqrt{3}-i)^{5}$$
View solution Problem 111
Consider two forces of equal magnitude acting on a point. (a) If the magnitude of the resultant is the sum of the magnitudes of the two forces, make a conjectur
View solution