Question

Let \(\alpha, \beta,\) and \(\gamma\) be the angles a vector \(\mathbf{v} \neq \mathbf{0}\) makes with the positive \(x, y,\) and \(z\) axes, respectively. Then \(\cos \alpha, \cos \beta,\) and \(\cos \gamma\) are called the direction cosines of the vector \(\mathbf{v}\). a. If \(\mathbf{v}=\left[\begin{array}{l}a \\ b \\ c\end{array}\right],\) show that \(\cos \alpha=\frac{a}{\|\mathbf{v}\|}, \cos \beta=\frac{b}{\|\mathbf{v}\|}\), and \(\cos \gamma=\frac{c}{\|\mathbf{v}\|}\). b. Show that \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1\).

Short answer

a. Direction cosines are \( \frac{a}{\|\mathbf{v}\|} \), \( \frac{b}{\|\mathbf{v}\|} \), \( \frac{c}{\|\mathbf{v}\|} \). b. Their squares sum to 1.

Step-by-step solution

Understand the Vector Components

Given a vector \( \mathbf{v} = \begin{bmatrix} a \ b \ c \end{bmatrix} \), its components along the \( x \)-, \( y \)-, and \( z \)-axes are \( a \), \( b \), and \( c \) respectively. The magnitude or norm of vector \( \mathbf{v} \) is denoted by \( \|\mathbf{v}\| \).

Calculate the Magnitude of the Vector

The magnitude of the vector \( \mathbf{v} \) is given by the formula: \( \|\mathbf{v}\| = \sqrt{a^2 + b^2 + c^2} \). This represents the length of the vector in the 3-dimensional space.

Define Direction Cosines

The direction cosines \( \cos \alpha \), \( \cos \beta \), and \( \cos \gamma \) for the vector \( \mathbf{v} \) are defined as \( \cos \alpha = \frac{a}{\|\mathbf{v}\|} \), \( \cos \beta = \frac{b}{\|\mathbf{v}\|} \), and \( \cos \gamma = \frac{c}{\|\mathbf{v}\|} \). This uses the standard geometric definition of cosine as adjacent (component) over hypotenuse (magnitude).

Verify the Expression for Direction Cosines

To check these expressions, recall that \( \cos \theta \) for any angle \( \theta \) is the adjacent side over the hypotenuse in a right triangle. Here, it is the component of the vector along the axis divided by its magnitude. Therefore, each direction cosine represents the proportion of the total vector along a particular axis.

Prove the Sum of Squares of Direction Cosines Equals One

We need to show that \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \). Substitute the direction cosines into the expression: \[ \left(\frac{a}{\|\mathbf{v}\|}\right)^2 + \left(\frac{b}{\|\mathbf{v}\|}\right)^2 + \left(\frac{c}{\|\mathbf{v}\|}\right)^2 = \frac{a^2}{a^2 + b^2 + c^2} + \frac{b^2}{a^2 + b^2 + c^2} + \frac{c^2}{a^2 + b^2 + c^2} \].

Simplify the Expression

Notice that the expression \( \frac{a^2}{a^2 + b^2 + c^2} + \frac{b^2}{a^2 + b^2 + c^2} + \frac{c^2}{a^2 + b^2 + c^2} \) simplifies to \( \frac{a^2 + b^2 + c^2}{a^2 + b^2 + c^2} = 1 \), proving that \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \).

Key concepts

Vector Components

Vectors are fundamental entities in physics and engineering that have both magnitude and direction. A key aspect of vectors is understanding their components, particularly in three-dimensional space. For a vector \( \mathbf{v} = \begin{bmatrix} a \ b \ c \end{bmatrix} \), the values \( a \), \( b \), and \( c \) are known as the vector components. These components represent the vector's projection or influence along each of the coordinate axes: \( x \)-axis, \( y \)-axis, and \( z \)-axis.

In simpler terms, the vector components tell you how much the vector moves in each direction. For instance, \( a \) represents the movement parallel to the \( x \)-axis. Similarly, \( b \) is the movement parallel to the \( y \)-axis, and \( c \) is for the \( z \)-axis. Understanding these components is crucial because they allow you to break down complex vector operations into simpler, one-dimensional operations along each axis.

• Vector components express how much of the vector points towards each axis.
• They help simplify calculations by breaking the vector into manageable parts.

Magnitude of a Vector

The magnitude of a vector, often termed as the vector's length, is a measure of how long the vector is. For a given vector \( \mathbf{v} = \begin{bmatrix} a \ b \ c \end{bmatrix} \), the magnitude is calculated using the formula: \[\|\mathbf{v}\| = \sqrt{a^2 + b^2 + c^2}\]This equation comes from the Pythagorean theorem applied in three dimensions. It essentially creates a "hypotenuse" out of the vector components.

Picture this: you have a vector represented by its components \( a \), \( b \), and \( c \). The magnitude is like stretching a straight line from the origin in space to the tip of the vector — giving the full length of this line. Understanding the magnitude is critical as it succinctly provides the vector's overall "size" or extent.

• The magnitude is the length or size of the vector.
• It is always a non-negative value.
• It combines components into a single measure of vector length.

Sum of Squares Identity

A fascinating result arises from the relationship between the vector components and their direction cosines. The sum of squares identity states that for direction cosines \( \cos \alpha \), \( \cos \beta \), and \( \cos \gamma \) derived from \( \cos \alpha = \frac{a}{\|\mathbf{v}\|} \), \( \cos \beta = \frac{b}{\|\mathbf{v}\|} \), \( \cos \gamma = \frac{c}{\|\mathbf{v}\|} \), we have: \[\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\]This identity reveals a powerful geometric property. It effectively communicates that the entire "directional energy" of the vector is perfectly partitioned among its three-dimensional components.

This is akin to saying that the contribution each direction makes to the vector doesn't exceed or fall short when combined — they form a perfect unit circle or sphere in three dimensions. What makes this identity particularly important is that it is universal, holding true for any non-zero vector, providing a consistent check on vector calculations.

• The identity indicates a balance in directionality across vector components.
• It's a reflection of basic trigonometric principles applied to vectors.
• Ensures a vector's direction cosines sum to unity.