Problem 32 - Precalculus

Question

Determine whether each pair of vectors is orthogonal. $$\langle 12,9\rangle \text { and }\langle 3,-4\rangle$$

Step-by-Step Solution

Answer

The vectors are orthogonal.

Step 1: Understand What Orthogonality Means

Two vectors are orthogonal to each other if their dot product is equal to zero. So, to determine if two vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) are orthogonal, we will calculate their dot product and check if it equals zero.

Step 2: Calculate the Dot Product

The dot product of two vectors \( \langle 12, 9 \rangle \) and \( \langle 3, -4 \rangle \) is calculated by multiplying the components and then adding the results. \((12 \times 3) + (9 \times -4) = 36 + (-36) = 0\)

Step 3: Check the Result

Since the dot product of the vectors \( \langle 12, 9 \rangle \) and \( \langle 3, -4 \rangle \) is 0, this confirms that the vectors are orthogonal to each other.

Key Concepts

Dot Product

The dot product is a fundamental concept in vector mathematics, often used to determine if two vectors are perpendicular or to find the projection of one vector onto another. The calculation involves taking two vectors, say \( \langle a, b \rangle \) and \( \langle c, d \rangle \), and multiplying their corresponding components, then adding the results together: \[ a \times c + b \times d \] In our example, to find the dot product of vectors \( \langle 12, 9 \rangle \) and \( \langle 3, -4 \rangle \), we perform the following operations:

• Multiply the first components: \( 12 \times 3 = 36 \)
• Multiply the second components: \( 9 \times -4 = -36 \)

Now, we sum the results of these multiplications: \( 36 + (-36) = 0 \). The final result of the dot product is 0, a crucial result that informs us about the relationship between the vectors.

Vector Components

Vectors are quantities possessing both magnitude and direction, and can be described in terms of their components. In two dimensions, a vector can be expressed as \( \langle a, b \rangle \), where \( a \) and \( b \) represent the horizontal and vertical components respectively. These components define the vector's position and direction in space, making it possible to carry out operations like addition, subtraction, and importantly, the dot product.

• The first coordinate \( a \) influences how far the vector extends horizontally.
• The second coordinate \( b \) impacts its vertical reach.

The components are essential in any vector operation because they allow us to break down and examine vectors in a straightforward manner. This breakdown facilitates easy calculation and understanding of more complex vector operations like determining orthogonality through the dot product.

Orthogonality Condition

Two vectors are orthogonal if they are perpendicular to each other in space. This is determined by the dot product of the vectors equalling zero. Orthogonality can have various practical meanings, such as denoting directions of maximum separation or independence. For example, the vectors \( \langle 12, 9 \rangle \) and \( \langle 3, -4 \rangle \) were calculated to have a dot product of 0: \[ (12 \times 3) + (9 \times -4) = 36 - 36 = 0 \] Since the result is zero, we conclude that the two vectors are indeed orthogonal.

• An orthogonal set of vectors can serve as a basis for vector spaces.
• In two and three dimensions, orthogonality simply means the vectors are at right angles to one another.

Recognizing orthogonal vectors is a key competence in vector analysis, as it simplifies many calculations and provides insight into the geometric relationships between vectors.