Problem 15 - University Calculus: Early Transcendentals

Question

Direction angles and direction cosines The direction angles \(\alpha, \beta,\) and \(\gamma\) of a vector \(\mathbf{v}=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}\) are defined as follows: \(\alpha\) is the angle between \(\mathbf{v}\) and the positive \(x\) -axis \((0 \leq \alpha \leq \pi)\) \(\beta\) is the angle between \(v\) and the positive \(y\) -axis \((0 \leq \beta \leq \pi)\) \(\gamma\) is the angle between \(\mathbf{v}\) and the positive \(z\) -axis \((0 \leq \gamma \leq \pi)\) (graph cannot copy) a. Show that $$ \cos \alpha=\frac{a}{|\mathbf{v}|}, \quad \cos \beta=\frac{b}{|\mathbf{v}|}, \quad \cos \gamma=\frac{c}{|\mathbf{v}|} $$ and \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 .\) These cosines are called the direction cosines of \(\mathbf{v}\). b. Unit vectors are built from direction cosines Show that if \(\mathbf{v}=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}\) is a unit vector, then \(a, b,\) and \(c\) are the direction cosines of \(\mathbf{v}\)

Step-by-Step Solution

Answer

Direction cosines are projections of \( \mathbf{v} \) on axes, divided by magnitude. For a unit vector, they are the components themselves.

Step 1: Magnitude of Vector v

Calculate the magnitude of the vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \). The magnitude is given by \( |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \).

Step 2: Expressing Direction Cosines

The direction cosines are calculated by projecting the vector onto the axes and dividing by its magnitude for normalization. Thus, the direction cosines are given by: \( \cos \alpha = \frac{a}{|\mathbf{v}|} \), \( \cos \beta = \frac{b}{|\mathbf{v}|} \), and \( \cos \gamma = \frac{c}{|\mathbf{v}|} \).

Step 3: Proving Sum of Squares of Direction Cosines Equals One

To show that the sum of the squares of the direction cosines equals one, compute: \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = \left( \frac{a}{|\mathbf{v}|} \right)^2 + \left( \frac{b}{|\mathbf{v}|} \right)^2 + \left( \frac{c}{|\mathbf{v}|} \right)^2 \). Simplify this to \( \frac{a^2 + b^2 + c^2}{(a^2 + b^2 + c^2)} = 1 \). This proves the identity.

Step 4: Understanding Unit Vectors and Direction Cosines

If \( \mathbf{v} \) is a unit vector, then \( |\mathbf{v}| = 1 \). Therefore, the direction cosines become \( \cos \alpha = a \), \( \cos \beta = b \), and \( \cos \gamma = c \), indicating that \( a, b, \) and \( c \) coincidentally are the direction cosines.

Key Concepts

Direction Cosines

Direction cosines are the angles a vector makes with the coordinate axes, which help us understand its orientation in space. For a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \), the direction cosines refer to the cosines of the angles formed between the vector and the axis directions.

• Cosine of the angle with the x-axis \( (\alpha) \): \( \cos \alpha = \frac{a}{|\mathbf{v}|} \)
• Cosine of the angle with the y-axis \( (\beta) \): \( \cos \beta = \frac{b}{|\mathbf{v}|} \)
• Cosine of the angle with the z-axis \( (\gamma) \): \( \cos \gamma = \frac{c}{|\mathbf{v}|} \)

The formula for direction cosines shows that each component of the vector is scaled down by its magnitude. This ensures the values range between -1 and 1. A fascinating property of direction cosines is that the sum of their squares is always equal to one:\[\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\].This relationship confirms the geometric nature of the vector and its balanced projection on all three axes.

Unit Vectors

A unit vector has a magnitude of one and is used to indicate direction. It is a scalar multiple of any vector that retains the same direction as the original vector but has a different magnitude. In mathematical terms, a unit vector is represented as:\[\mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|}\]In the case where the vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \) is already a unit vector, its magnitude, \(|\mathbf{v}|\), is one. This implies:

• The direction cosines transform directly to the components: \( a = \cos \alpha \),
• \( b = \cos \beta \),
• \( c = \cos \gamma \).

Unit vectors are essential for defining directions without concern for magnitude, aiding in creating clean and scalable mathematical models.

Magnitude of Vector

The magnitude of a vector, often referred to as its 'length', provides a measure of how long a vector is, irrespective of the direction in which it points. For any vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \), the magnitude is derived by applying the Pythagorean theorem. The formula for finding the magnitude is:\[|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}\]Understanding the magnitude is crucial as it scales the direction cosines, transforming the components into a normalized form. The magnitude also ensures that vector calculations align with geometric interpretations in real, physical space, acting as a bridge between abstract mathematical concepts and tangible application.

Vector Components

Vector components are the projections of a vector along the coordinate axes. For the vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \), each axis component represents how much of the vector's magnitude is directed along that particular axis.

• Component along the x-axis: The term \( a \mathbf{i} \) illustrates the extent of \( \mathbf{v} \) along the x-direction.
• Component along the y-axis: The term \( b \mathbf{j} \) captures the amount of \( \mathbf{v} \) extending along the y-direction.
• Component along the z-axis: \( c \mathbf{k} \) quantifies \( \mathbf{v} \)'s elongation along the z-direction.

These components make it easier to manipulate and work with vectors in calculations, providing clarity on the vector's influence in multi-dimensional space. In vector addition, subtraction, or when constructing vector expressions, focusing on components streamlines the process.