Problem 30 - Precalculus : Building Concepts and Connections

Question

Determine whether the given pairs of vectors are orthogonal. $$\mathbf{v}=\langle 3,5\rangle, \mathbf{w}=\left\langle\frac{5}{6}, \frac{1}{2}\right\rangle$$

Step-by-Step Solution

Answer

The given vectors are not orthogonal because their dot product is not zero, but 5.

Step 1: Write down the dot product formula

Start by recalling that the dot product of two vectors \(\mathbf{v} = \langle v_1,v_2 \rangle\) and \(\mathbf{w} = \langle w_1,w_2 \rangle\) is given by \(\mathbf{v} \cdot \mathbf{w} = v_1w_1 + v_2w_2\).

Step 2: Substitute the given vectors into the formula

Now substitute the given vectors into the dot product formula: \( \mathbf{v} \cdot \mathbf{w} = 3 \cdot \frac{5}{6} + 5 \cdot \frac{1}{2}\).

Step 3: Compute the dot product

Simplify the expression to calculate the dot product: \( \mathbf{v} \cdot \mathbf{w} = \frac{15}{6} + \frac{5}{2} = \frac{5}{2} + \frac{5}{2} = 5\).

Step 4: Check if the dot product is zero

Vectors are orthogonal if their dot product equals zero. Here, the dot product is 5, not zero. Hence, the given vectors are not orthogonal.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation with vectors in mathematics. It allows us to multiply two vectors and get a scalar (a single number) as a result. When dealing with two-dimensional vectors like \( \mathbf{v} = \langle v_1, v_2 \rangle \) and \( \mathbf{w} = \langle w_1, w_2 \rangle \), the dot product is computed by multiplying their respective components and adding the results together. This is expressed as:

• \( \mathbf{v} \cdot \mathbf{w} = v_1w_1 + v_2w_2 \)

In the original exercise, this formula was used to determine if the vectors \( \mathbf{v} = \langle 3,5 \rangle \) and \( \mathbf{w} = \langle \frac{5}{6}, \frac{1}{2} \rangle \) were orthogonal. After substituting and calculating, they found a result of 5. The vectors were therefore not orthogonal as orthogonality means a dot product result of zero.
This method is handy in many applications, such as calculating angles between vectors or checking if they're perpendicular.

Vectors in Mathematics

Vectors are quantities that have both a magnitude (size) and a direction. They can be represented in different dimensions, but for simplicity, we'll focus on two-dimensional vectors like those used in the original exercise. Vectors in mathematics are defined with components, usually expressed in the form \( \mathbf{v} = \langle v_1, v_2 \rangle \).
They are used to describe anything involving both direction and magnitude, such as:

• Force
• Velocity
• Displacement

When adding or subtracting vectors, it's essential to perform the operations component-wise, calculating each dimension separately. For example, if we have vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), their addition would be:

• \( \mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2 \rangle \)

This way of working with vectors ensures we accurately represent both the magnitude and direction of combined forces or movements.

Vector Operations

Vector operations refer to various mathematical actions we can perform using vectors. These operations allow us to manipulate and extract useful information from vectors in mathematics. In the original exercise, we computed the dot product, which is critical for several vector calculations. Other common vector operations include:

• **Addition**: Adding corresponding components of two vectors, as in \( \mathbf{v} + \mathbf{w} = \langle v_1 + w_1, v_2 + w_2 \rangle \)
• **Subtraction**: Subtracting corresponding components, such as \( \mathbf{v} - \mathbf{w} = \langle v_1 - w_1, v_2 - w_2 \rangle \)
• **Scalar Multiplication**: Multiplying each component by a scalar (single number), e.g., \( c\mathbf{v} = \langle cv_1, cv_2 \rangle \)
• **Magnitude**: Calculating the length or size of a vector using the formula \( \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2} \)

These operations are important for exhaustive analysis, enabling detailed insights into various phenomena. They are extensively used in geometric calculations, physics simulations, and computer graphics.