Problem 15 - Advanced Engineering Mathematics

Question

Determine which pairs of the following vectors are orthogonal: (a) \(\langle 2,0,1\rangle\) (b) \(3 \mathbf{i}+2 \mathbf{j}-\mathbf{k}\) (c) \(2 \mathbf{i}-\mathbf{j}-\mathbf{k}\) (d) \(\mathbf{i}-4 \mathbf{j}+6 \mathbf{k}\) (e) \(\langle 1,-1,1\rangle\) (f) \(\langle-4,3,8\rangle\)

Step-by-Step Solution

Answer

Pairs (a,f), (b,e), and (c,d) are orthogonal.

Step 1: Understand Orthogonality

Two vectors are orthogonal if their dot product is zero. To check orthogonality between any two vectors, calculate their dot product and see if the result is zero.

Step 2: Express All Vectors in Component Form

Rewrite all vectors properly in component form:(a) \( \langle 2, 0, 1 \rangle \)(b) \( \langle 3, 2, -1 \rangle \)(c) \( \langle 2, -1, -1 \rangle \)(d) \( \langle 1, -4, 6 \rangle \)(e) \( \langle 1, -1, 1 \rangle \)(f) \( \langle -4, 3, 8 \rangle \)

Step 3: Pair the Vectors for Checking

Create all possible pairs of vectors from the list: (a,b), (a,c), (a,d), (a,e), (a,f), (b,c), (b,d), (b,e), (b,f), (c,d), (c,e), (c,f), (d,e), (d,f), and (e,f).

Step 4: Calculate Dot Product for Each Pair

Find the dot product for each pair of vectors:- (a,b): \( 2 \times 3 + 0 \times 2 + 1 \times (-1) = 6 + 0 - 1 = 5 \)- (a,c): \( 2 \times 2 + 0 \times (-1) + 1 \times (-1) = 4 + 0 - 1 = 3 \)- (a,d): \( 2 \times 1 + 0 \times (-4) + 1 \times 6 = 2 + 0 + 6 = 8 \)- (a,e): \( 2 \times 1 + 0 \times (-1) + 1 \times 1 = 2 + 0 + 1 = 3 \)- (a,f): \( 2 \times (-4) + 0 \times 3 + 1 \times 8 = -8 + 0 + 8 = 0 \)- (b,c): \( 3 \times 2 + 2 \times (-1) + (-1) \times (-1) = 6 - 2 + 1 = 5 \)- (b,d): \( 3 \times 1 + 2 \times (-4) + (-1) \times 6 = 3 - 8 - 6 = -11 \)- (b,e): \( 3 \times 1 + 2 \times (-1) + (-1) \times 1 = 3 - 2 - 1 = 0 \)- (b,f): \( 3 \times (-4) + 2 \times 3 + (-1) \times 8 = -12 + 6 - 8 = -14 \)- (c,d): \( 2 \times 1 + (-1) \times (-4) + (-1) \times 6 = 2 + 4 - 6 = 0 \)- (c,e): \( 2 \times 1 + (-1) \times (-1) + (-1) \times 1 = 2 + 1 - 1 = 2 \)- (c,f): \( 2 \times (-4) + (-1) \times 3 + (-1) \times 8 = -8 - 3 - 8 = -19 \)- (d,e): \( 1 \times 1 + (-4) \times (-1) + 6 \times 1 = 1 + 4 + 6 = 11 \)- (d,f): \( 1 \times (-4) + (-4) \times 3 + 6 \times 8 = -4 -12 + 48 = 32 \)- (e,f): \( 1 \times (-4) + (-1) \times 3 + 1 \times 8 = -4 - 3 + 8 = 1 \)

Step 5: Identify Orthogonal Pairs

The pairs that result in a dot product of zero are orthogonal: - The pair (a,f) with dot product 0. - The pair (b,e) with dot product 0. - The pair (c,d) with dot product 0.

Key Concepts

Dot Product

The dot product is a fundamental concept when dealing with vector mathematics, especially for those studying orthogonal vectors. It refers to the product of two vectors and gives a scalar result. Mathematically, if you have two vectors \( \mathbf{A} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{B} = \langle b_1, b_2, b_3 \rangle \), the dot product is calculated as:
\[ \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \]
This operation essentially measures how much one vector extends in the direction of another. Its value can tell us a lot about the relationship between the two vectors. For instance:

• If the dot product is zero, the vectors are orthogonal, or at right angles to each other.
• If the dot product is positive, the vectors point in a similar direction.
• If the dot product is negative, the vectors point in opposite directions.

Vector Components

Understanding vector components is crucial when working with vectors. In physics and mathematics, a vector is often expressed in component form, making calculations easier. Each vector is represented by its projections along the respective axes, typically written as \( \langle x, y, z \rangle \) for three-dimensional space.
Breaking down a vector into components helps in understanding its precise direction and magnitude:

• The x-component shows how far the vector goes along the x-axis.
• The y-component shows how far it extends along the y-axis.
• The z-component indicates the vector's projection along the z-axis.

For example, a vector \( \langle 2, 0, 1 \rangle \) means it moves 2 units along the x-axis, 0 units along the y-axis, and 1 unit along the z-axis. Knowing these components enables you to calculate the dot product, find the angle between vectors, and determine orthogonality.

Vector Pairs

When considering pairs of vectors, it is imperative to recognize the impact that these pairs have when calculating their interactions, such as their dot product. Pairing vectors systematically allows us to inspect all possible interactions between vectors in a set.
For any exercise involving vector pairs, make sure you:

• List all possible pairs: This includes all possible combinations without repetition. For six vectors, create pairs like (a, b), (a, c), and so on.
• Calculate each pair's dot product: Knowing whether their dot product is zero tells you if they are orthogonal.

Understanding the significance of vector pairs becomes clear when assessing orthogonality among multiple vectors. By calculating and comparing, we can quickly determine which relationship holds, ultimately solving the problem at hand.

Calculating Orthogonality

Orthogonality refers to the concept where two vectors are perpendicular to each other. To verify this relationship, calculate the dot product of the two vectors. If the result is zero, the vectors are orthogonal—a critical property in various applications such as computer graphics and physics.
Here’s how you determine if vectors are orthogonal:

• Convert all vectors into component form if they aren't already.
• Use the dot product formula for each vector pair.
• Look for pairs where the dot product equals zero.

For exercises, you may follow predefined vector pairs to verify where orthogonality holds. Such calculations include: - Pair (a, f) in our example yielded a dot product of 0. - Similarly, pairs (b, e) and (c, d) were also found to be orthogonal.
This systematic approach ensures that you accurately identify orthogonal vector pairs, helping deepen your understanding of vector relationships.