Problem 30 - Algebra and Trigonometry

Question

Use the dot product to determine whether v and w are orthogonal. $$ \mathbf{v}=5 \mathbf{i}, \quad \mathbf{w}=-6 \mathbf{i} $$

Step-by-Step Solution

Answer

No, the vectors \( \mathbf{v} \) and \( \mathbf{w} \) are not orthogonal.

Step 1: Identify the vectors

The vectors given are \( \mathbf{v}=5 \mathbf{i} \) and \( \mathbf{w}=-6 \mathbf{i} \)

Step 2: Compute the dot product

The dot product of two vectors \( \mathbf{v} \) and \( \mathbf{w} \) is calculated by multiplying the corresponding i, j, and k coefficients. In this case, only i coefficients are given. So, the dot product is \( (5)(-6) = -30 \).

Step 3: Determine orthogonality

Since the dot product of vectors \( \mathbf{v} \) and \( \mathbf{w} \) is not zero (-30 ≠ 0), the vectors are not orthogonal.

Key Concepts

Orthogonal Vectors

In vector mathematics, two vectors are said to be orthogonal if they meet at a right angle (90 degrees). The concept of orthogonal vectors is crucial because it simplifies many mathematical operations, particularly those involving geometry and physics.
For two vectors to be orthogonal, their dot product must be zero. The dot product gives us a measure of how much one vector extends in the direction of another. If there's no extension at all, as would be the case when vectors are perpendicular, the dot product is zero.
In the exercise provided, the vectors \( \mathbf{v}=5 \mathbf{i} \) and \( \mathbf{w}=-6 \mathbf{i} \) were considered. By calculating the dot product and finding it was \(-30\), it became clear these vectors are not orthogonal. This is because their product wasn’t zero, indicating they do not meet at a right angle.

Vector Mathematics

Vector mathematics involves operations that are fundamental to physics and engineering. Vectors can represent quantities like velocity, force, and displacement. These quantities have both magnitude and direction, which differentiates vectors from scalars.
The exercise demonstrates the use of the dot product. This operation is one way to measure how two vectors relate in space. Calculating the dot product of vectors \( \mathbf{v}=5 \mathbf{i} \) and \( \mathbf{w}=-6 \mathbf{i} \) involves multiplying their components. Since these vectors only include the \( i \)-component, the calculation simplifies to \( 5 \times (-6) = -30 \).
This highlights how simple vector calculations can be when vectors are aligned along the same axis. More complex vectors have components in multiple directions, each requiring consideration. Nevertheless, the intuitive result that only non-zero products prevent orthogonality simplifies reasoning about vector orientation.

Algebra

In algebra, the dot product is an example of how abstract mathematical concepts can be applied to solve problems in physics and engineering. Algebra provides the tools for working with vectors and understanding relationships between their components.
When using algebra to solve for the dot product between \( \mathbf{v} \) and \( \mathbf{w} \), do the following:

• Identify the corresponding components. Here, that is 5 for \( \mathbf{v} \) and \(-6\) for \( \mathbf{w} \).
• Multiply these components: \( 5 \times -6 \).
• Summing the products of all corresponding components gives the dot product, which here is \(-30\).

This algebraic operation shows how the orientation and magnitude of vectors directly affect their relationships. Vectors that are not orthogonal have a non-zero dot product, reflecting the presence of an angle less than or greater than 90 degrees between them.