Problem 29 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

The given vectors \(\mathbf{v}=\left\langle\frac{1}{3}, 2\right\rangle\) and \(\mathbf{w}=\left\langle 6, \frac{5}{2}\right\rangle\) are not orthogonal.

Step 1: Identify the given vectors

The given vectors are \(\mathbf{v}=\left\langle\frac{1}{3}, 2\right\rangle\) and \(\mathbf{w}=\left\langle 6, \frac{5}{2}\right\rangle\).

Step 2: Calculate the dot product

The dot product of two vectors \(\mathbf{a} = \langle a_1, a_2 \rangle\) and \(\mathbf{b} = \langle b_1, b_2 \rangle\) is calculated as \(\mathbf{a} \cdot \mathbf{b} = a_1*b_1 + a_2*b_2\). Hence, the dot product for \(\mathbf{v} \) and \(\mathbf{w}\) is \(\left(\frac{1}{3} * 6\right) + \left(2 * \frac{5}{2}\right)\) which simplifies to \(2 + 5 = 7\).

Step 3: Determine orthogonality

The vectors \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal if and only if their dot product is zero. Since the dot product here is 7, which is not equal to zero, the vectors are not orthogonal.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector mathematics that combines two vectors to result in a scalar. To calculate the dot product of two-dimensional vectors, you multiply the corresponding components from each vector and add the products together. Mathematically, if you have two vectors, \(\mathbf{a} = \langle a_1, a_2 \rangle\) and \(\mathbf{b} = \langle b_1, b_2 \rangle\), their dot product is \(\mathbf{a} \cdot \mathbf{b} = a_1\cdot b_1 + a_2\cdot b_2\).

This computation is crucial because it can reveal information about the angle between the vectors. For instance, if the dot product equals zero, it indicates that the vectors are orthogonal (perpendicular) to each other. However, a non-zero dot product means that they are not orthogonal and form some other angle relative to each other.

Vector Orthogonality

Vector orthogonality is a concept referring to vectors being at a right angle to each other. This is a significant geometric and algebraic property in vector space. The ultimate test to determine if two vectors are orthogonal is to compute their dot product. If the result is exactly zero, the vectors are orthogonal. This criterion applies regardless of the dimensionality of the vectors involved.

For illustrative purposes, in our exercise, when we calculated the dot product of vectors \(\mathbf{v}\) and \(\mathbf{w}\) as 7, this immediately tells us that the vectors are not orthogonal. It's important to note that orthogonality in higher dimensions follows the same rule—the dot product must be zero.

Vector Operations

Vector operations encompass several basic actions that can be performed with vectors, including addition, subtraction, scalar multiplication, and the dot product. To add or subtract vectors, you simply add or subtract their corresponding components. Scalar multiplication involves multiplying each component of the vector by a scalar, or real number, resulting in a new vector.

In order to provide a comprehensive understanding, let's focus on vector addition. When adding two vectors, such as \(\mathbf{u} = \langle u_1, u_2 \rangle\) and \(\mathbf{v} = \langle v_1, v_2 \rangle\), you'll obtain \(\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle\). These operations are essential when dealing with forces, velocities, or any physical quantity that has both magnitude and direction.