Problem 30 - Advanced Engineering Mathematics

Question

Show that if two nonzero vectors \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal, then their direction cosines satisfy $$ \cos \alpha_{1} \cos \alpha_{2}+\cos \beta_{1} \cos \beta_{2}+\cos \gamma_{1} \cos \gamma_{2}=0 $$

Step-by-Step Solution

Answer

The direction cosines satisfy: \(\cos \alpha_1 \cos \alpha_2 + \cos \beta_1 \cos \beta_2 + \cos \gamma_1 \cos \gamma_2 = 0\).

Step 1: Understand Orthogonality of Vectors

Orthogonal vectors are vectors that are perpendicular to each other. If two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal, their dot product is zero, i.e., \(\mathbf{a} \cdot \mathbf{b} = 0\). This is because the cosine of the angle between the vectors is zero when the angle is 90 degrees.

Step 2: Express Vectors Using Direction Cosines

Every vector can be expressed in terms of its component directions. For vector \(\mathbf{a}\), \(\cos \alpha_1, \cos \beta_1, \cos \gamma_1\) are the direction cosines with respect to the x, y, and z axes respectively. Thus, \(\mathbf{a} = a(\cos \alpha_1, \cos \beta_1, \cos \gamma_1)\) and similarly, \(\mathbf{b} = b(\cos \alpha_2, \cos \beta_2, \cos \gamma_2)\).

Step 3: Compute the Dot Product Using Direction Cosines

The dot product \(\mathbf{a} \cdot \mathbf{b}\) is computed as:\[a \cos \alpha_1 \cdot b \cos \alpha_2 + a \cos \beta_1 \cdot b \cos \beta_2 + a \cos \gamma_1 \cdot b \cos \gamma_2 = 0\]This simplifies to:\[ab(\cos \alpha_1 \cos \alpha_2 + \cos \beta_1 \cos \beta_2 + \cos \gamma_1 \cos \gamma_2) = 0\]

Step 4: Simplify and Cancel Non-zero Scalars

Since \(a\) and \(b\) are non-zero scalars, we can divide through by \(ab\) to simplify the equation while maintaining equality:\[\cos \alpha_1 \cos \alpha_2 + \cos \beta_1 \cos \beta_2 + \cos \gamma_1 \cos \gamma_2 = 0\]Thus, showing the relationship between the direction cosines of two orthogonal vectors.

Key Concepts

Direction Cosines

Direction cosines are a way to represent a vector in 3D space concerning the coordinate axes. Specifically, for any vector, direction cosines help us understand how much the vector is aligned with each axis.
For a vector \( \mathbf{a} \), the direction cosines are represented as \( \cos \alpha, \cos \beta, \text{ and } \cos \gamma \) for the x, y, and z axes, respectively. These are essentially the cosines of the angles that the vector makes with each of the axes.
Every vector can be decomposed into:

• \( \mathbf{a} = a(\cos \alpha_1, \cos \beta_1, \cos \gamma_1) \)
• \( \mathbf{b} = b(\cos \alpha_2, \cos \beta_2, \cos \gamma_2) \)

Direction cosines are useful because they can tell us about the alignment and orientation of vectors concerning a set of reference axes, making them crucial for many applications in physics and engineering.

Dot Product

The dot product is a fundamental operation in vector algebra used to find the angle between two vectors or to check for orthogonality. When two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are multiplied using the dot product, the result is a scalar.Mathematically, the dot product is calculated as:\[ \mathbf{a} \cdot \mathbf{b} = a_xb_x + a_yb_y + a_zb_z \]where \( a_x, a_y, a_z \) are the components of vector \( \mathbf{a} \) and \( b_x, b_y, b_z \) are the components of vector \( \mathbf{b} \).

Using the direction cosines, the dot product also becomes:\[ \mathbf{a} \cdot \mathbf{b} = ab(\cos \alpha_1 \cos \alpha_2 + \cos \beta_1 \cos \beta_2 + \cos \gamma_1 \cos \gamma_2) \]
This form is particularly useful when studying properties like orthogonality because if the product simplifies to zero, the vectors are orthogonal.

Orthogonal Vectors

Orthogonal vectors are those that are perpendicular to each other. This means the angle between them is exactly 90 degrees. A key property of orthogonal vectors is that their dot product is zero.
This is because at 90 degrees, the cosine of the angle between the vectors is zero, which makes the entire product zero. For two vectors \( \mathbf{a} \) and \( \mathbf{b} \), checking for orthogonality involves computing:

• \( \mathbf{a} \cdot \mathbf{b} = 0 \) if they are orthogonal.

In community settings such as engineering or graphics, knowing that vectors are orthogonal can simplify calculations because perpendicular forces don't influence each other.