Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether u and v are orthogonal, parallel, or neither. $$\begin{aligned} &\mathbf{u}=-\frac{9}{10} \mathbf{i}+3 \mathbf{j}\\\ &\mathbf{v}=-5 \mathbf{i}+\frac{3}{2} \mathbf{j} \end{aligned}$$

Step-by-Step Solution

Answer

Vectors \( \mathbf{u} \) and \( \mathbf{v} \) are neither orthogonal nor parallel.

Step 1: Calculate the Dot Product

We calculate the dot product of vectors \( \mathbf{u} \) and \( \mathbf{v} \) using the formula \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \). Here, \( u_1 = -\frac{9}{10}, u_2 = 3, v_1 = -5, v_2 = \frac{3}{2} \). So, \( \mathbf{u} \cdot \mathbf{v} = (-\frac{9}{10} \times (-5)) + (3 \times \frac{3}{2}) = 4.5 + 4.5 = 9.

Step 2: Check for Orthogonality

Two vectors are orthogonal if their dot product equals zero. However in this case, \( \mathbf{u} \cdot \mathbf{v} = 9 \neq 0 \), so vectors \( \mathbf{u} \) and \( \mathbf{v} \) are not orthogonal.

Step 3: Check for Parallelism

Two vectors are parallel if one is a scalar multiple of the other. We divide the \( i \) component of vector \( \mathbf{v} \) by \( \mathbf{u} \)'s i component. Similarly, we divide \( j \) component of vector \( \mathbf{v} \) by \( \mathbf{u} \)'s j component. We get \( \frac{-5}{-\frac{9}{10}} = 5.56 \) and \( \frac{\frac{3}{2}}{3} = 0.5 \). The divisors are not equal. Hence, vectors \( \mathbf{u} \) and \( \mathbf{v} \) are not parallel.

Key Concepts

Orthogonal Vectors

Orthogonal vectors are two vectors that meet at a right angle. We say these vectors are perpendicular to each other. When two vectors are orthogonal, the angle between them is 90 degrees. This special property gives rise to an important mathematical condition:

• The dot product of two orthogonal vectors is zero.

In mathematics, the dot product is a way to understand if two vectors point in significantly different directions. For instance, consider vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \). The dot product is given by the formula \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \).
To check if two vectors are orthogonal, simply calculate their dot product. If the result is zero, they are orthogonal. Otherwise, they are not perpendicular to each other. In our problem, since the dot product is 9 and not zero, vectors \( \mathbf{u} \) and \( \mathbf{v} \) are not orthogonal.

Dot Product Calculation

The dot product is a powerful tool in vector analysis. It helps in understanding the relationship between two vectors. To compute the dot product of vectors, we multiply their corresponding components and add the results. Given two vectors, \( \mathbf{u} = u_1\mathbf{i} + u_2\mathbf{j} \) and \( \mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} \), the dot product is calculated as follows:

• Dot product: \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \)

In this exercise:

• \( u_1 = -\frac{9}{10} \), \( u_2 = 3 \)
• \( v_1 = -5 \), \( v_2 = \frac{3}{2} \)

The calculation thus becomes: \((-\frac{9}{10} \times (-5)) + (3 \times \frac{3}{2}) \)
This simplifies to: \( 4.5 + 4.5 = 9 \).
The dot product gives us insight. If it were zero, the vectors would be orthogonal. Since it's not, they are not orthogonal. The dot product also plays a part in testing for parallelism, as there's a potential link to angle calculation.

Parallel Vectors

Parallel vectors occur when two vectors point exactly in the same or opposite direction. You can think of them as being perfectly aligned along a single line, with possible different magnitudes. Two vectors are parallel if one is a scalar (a single number) multiple of the other.
To determine if vectors are parallel:

• Compute the ratio of corresponding components.
• If every pair of ratios is equal, the vectors are parallel.

Consider vectors \( \mathbf{u} \) and \( \mathbf{v} \), with i and j components. You calculate:

• \( \frac{v_1}{u_1} \) and \( \frac{v_2}{u_2} \)

For our vectors, these ratios are: \( \frac{-5}{-\frac{9}{10}} = 5.56 \) and \( \frac{\frac{3}{2}}{3} = 0.5 \).
Since these calculated values are not equivalent, the vectors are not scalar multiples of each other, meaning they are not parallel. When calculating parallelism, checking these ratios helps identify whether the vectors lie on the same line. In this instance, we've confirmed \( \mathbf{u} \) and \( \mathbf{v} \) aren't parallel by demonstrating these unexpected ratio values.