Problem 24 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Yes, the vectors v and w are orthogonal because their dot product equals zero.

Step 1: Express vectors in component form

Vector v can be represented as \(v = (1, 1)\), and vector w can be represented as \(w = (-1, 1)\). The component form is derived from the coefficients of unit vectors i and j.

Step 2: Calculate the dot product

The dot product of two vector v and w is obtained by multiplying corresponding components and adding the products. So, \(v \cdot w = v_{1} \cdot w_{1} + v_{2} \cdot w_{2} = (1 \cdot -1) + (1 \cdot 1) = -1 + 1 = 0\).

Step 3: Check for Orthogonality

Since the dot product of vectors v and w is 0, vectors v and w are orthogonal.

Key Concepts

Vector Components

Vectors are fundamental in representing quantities that have both a magnitude and a direction. To properly manipulate vectors in mathematical problems, it is crucial to express them in their component form. This involves breaking a vector down into its individual parts along the axes, typically represented by the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in a two-dimensional space. For example, consider the vectors \(\mathbf{v} = \mathbf{i} + \mathbf{j}\) and \(\mathbf{w} = -\mathbf{i} + \mathbf{j}\).
In component form, \(\mathbf{v}\) becomes \((1, 1)\), indicating 1 unit in the \(\mathbf{i}\) or x-direction and 1 unit in the \(\mathbf{j}\) or y-direction. Similarly, \(\mathbf{w}\) translates to \((-1, 1)\), showing a movement of -1 unit in the x-direction and 1 unit upwards. These simple numerical expressions make it easier to perform operations such as addition, subtraction, and particularly the dot product.

Orthogonal Vectors

Orthogonality in vectors is a critical concept in geometry and physics, indicating that two vectors are perpendicular to each other. This property is significant because, in many applications, orthogonal vectors can simplify analyses, computations, and interpretations of physical phenomena. To determine if two vectors are orthogonal, we use their dot product. The dot product of two orthogonal vectors is zero.
Applying this to our vectors \(\mathbf{v} = (1, 1)\) and \(\mathbf{w} = (-1, 1)\):

• Calculate the dot product: \(\mathbf{v} \cdot \mathbf{w} = (1)(-1) + (1)(1) = -1 + 1 = 0\).
• Since the dot product is zero, vectors \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal.

This result confirms they are perpendicular, which can be visualized as \(\mathbf{v}\) and \(\mathbf{w}\) forming a right angle at the intersection of an x-y coordinate plane.

Vector Multiplication

Vector multiplication is not as straightforward as multiplying ordinary numbers, as it involves more complex operations like the dot product and cross product. Here, we focus on the dot product, which is a fundamental operation providing insights into the relationship between two vectors.
The dot product, or scalar product, of two vectors \(\mathbf{a} = (a_1, a_2)\) and \(\mathbf{b} = (b_1, b_2)\), is calculated as:
\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2\]
Notice that this operation returns a scalar, not a vector. It's a measure of how much one vector extends in the direction of another. For instance, in the problem exercise, the dot product of \(\mathbf{v} = (1, 1)\) and \(\mathbf{w} = (-1, 1)\) was found to be:

• \(1 \times -1 + 1 \times 1 = 0\)

This confirms that when the dot product equals zero, the vectors are orthogonal, as there's no projection of one vector onto the other.