Problem 15 - Thomas Calculus

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: \begin{equation} \begin{array}{l}{\alpha \text { is the angle between } v \text { and the positive } x \text { -axis }(0 \leq \alpha \leq \pi)} \\ {\beta \text { is the angle between } v \text { and the positive } y \text { -axis }(0 \leq \beta \leq \pi)} \\ {\gamma \text { is the angle between } v \text { and the positive } z \text { -axis }(0 \leq \gamma \leq \pi)}\end{array} \end{equation} a. Show that \begin{equation} \cos \alpha=\frac{a}{|\mathbf{v}|}, \quad \cos \beta=\frac{b}{|\mathbf{v}|}, \quad \cos \gamma=\frac{c}{|\mathbf{v}|} \end{equation} \begin{equation} \begin{array}{l}{\text { and } \cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 . \text { These cosines are called }} \\ {\text { the direction cosines of } \mathbf{v} .}\end{array} \end{equation} b. Unit vectors are built from direction cosines Show that if \(\quad \mathbf{v}=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}\) is a unit vector, then \(a, b,\) and \(c\) are the direction cosines of \(\mathbf{v} .\)

Step-by-Step Solution

Answer

Direction cosines are ratios of each component to the vector magnitude, and for unit vectors, they are the components themselves.

Step 1: Define Vector Magnitude

The magnitude of the vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \) is given by the formula \( |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \). This step is necessary to find the direction cosines.

Step 2: Express Direction Cosines

The direction angles \( \alpha, \beta, \gamma \) are the angles between \( \mathbf{v} \) and the positive \( x, y, \) and \( z \) axes, respectively. Thus, the direction cosines are given by \( \cos \alpha = \frac{a}{|\mathbf{v}|} \), \( \cos \beta = \frac{b}{|\mathbf{v}|} \), and \( \cos \gamma = \frac{c}{|\mathbf{v}|} \). This is because each cosine is the component of the vector divided by its magnitude.

Step 3: Derive the Direction Cosine Identity

Using the expressions for the direction cosines, show that the sum of their squares equals 1: \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = \left(\frac{a}{|\mathbf{v}|}\right)^2 + \left(\frac{b}{|\mathbf{v}|}\right)^2 + \left(\frac{c}{|\mathbf{v}|}\right)^2 = \frac{a^2 + b^2 + c^2}{|\mathbf{v}|^2} = 1 \), since \( a^2 + b^2 + c^2 = |\mathbf{v}|^2 \).

Step 4: Check Unit Vector Condition

If \( \mathbf{v} \) is a unit vector, then \( |\mathbf{v}| = 1 \). Thus, \( a^2 + b^2 + c^2 = 1^2 = 1 \), making \( a, b, c \) the direction cosines themselves, as \( \cos \alpha = a, \cos \beta = b, \cos \gamma = c \).

Key Concepts

Direction Angles

Direction angles are crucial when working with vectors as they relate the vector's orientation to the coordinate axes. A direction angle is essentially the measure between a vector and one of the coordinate axes, giving us insight into its orientation.

For a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \), the direction angles are defined as follows:

• \( \alpha \) is the angle between \( \mathbf{v} \) and the positive \( x \)-axis.
• \( \beta \) is the angle between \( \mathbf{v} \) and the positive \( y \)-axis.
• \( \gamma \) is the angle between \( \mathbf{v} \) and the positive \( z \)-axis.

These angles are constrained such that \( 0 \leq \alpha, \beta, \text{ and } \gamma \leq \pi \). Recognizing these directional relationships facilitates the calculation of direction cosines which help represent the vector in a more standard form.

Vector Magnitude

The magnitude of a vector, often called the vector length or norm, is a measure of how long the vector is. It is calculated using the components of the vector.

For a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \), its magnitude is expressed mathematically as:
\[| \mathbf{v}| = \sqrt{a^2 + b^2 + c^2}.\]Calculating the magnitude is crucial for finding direction cosines because these cosines are essentially the normalized components of the vector.

Without the magnitude, we cannot properly scale the components \( a, b, \text{ and } c \), which are necessary for calculating their respective angles with the coordinate axes.

Unit Vectors

Unit vectors are an integral part of vector mathematics and are vectors with a magnitude of one. Due to this property, they are primarily used to specify directions and are often derived from other vectors by dividing the vector by its magnitude.

A unit vector \( \mathbf{u} \) in the direction of a vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \) can be represented as:
\[\mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|} = \frac{a}{|\mathbf{v}|} \mathbf{i} + \frac{b}{|\mathbf{v}|} \mathbf{j} + \frac{c}{|\mathbf{v}|} \mathbf{k}.\]When \( \mathbf{v} \) is already a unit vector, this simplifies further. Its components, \( a, b, \text{ and } c \), become the direction cosines directly, because the magnitude \(|\mathbf{v}|\) equals 1.

Thus, direction cosines provide a direct measure of the vector's alignment along the axes in its smallest form, which is incredibly useful for expressing directions succinctly and uniformly.