Problem 57 - Precalculus with Limits

Question

Determining Orthogonal Vectors In Exercises \(53-58\) , determine whether \(u\) and \(v\) are orthogonal. $$\mathbf{u}=2 \mathbf{i}-2 \mathbf{j}$$ $$\mathbf{v}=-\mathbf{i}-\mathbf{j}$$

Step-by-Step Solution

Answer

Yes, the vectors \(\mathbf{u}=2 \mathbf{i}-2 \mathbf{j}\) , and \(\mathbf{v}=-\mathbf{i}-\mathbf{j}\) are orthogonal because their dot product is zero.

Step 1: Determine the Vectors

First, identify the given vectors which are; \(\mathbf{u}=2 \mathbf{i}-2 \mathbf{j}\) , and \(\mathbf{v}=-\mathbf{i}-\mathbf{j}\)

Step 2: Calculate the Dot Product

The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is found by multiplying corresponding components and then summing all. In this case, multiply \(i\)-components together, \(j\)-components together and \(j\)-components together, then sum these three products. The formula for dot product is \( \mathbf{u} . \mathbf{v} = (2)(-1) + (-2)(-1)\)

Step 3: Determine if Orthogonal

The vectors are orthogonal if the dot product is zero. In the precedent step, we found that the dot product is equal to 0. Therefore, \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics that often helps us to determine relationships between vectors, such as orthogonality. It involves multiplying corresponding components of two vectors and then summing these products.

For vectors in the plane, like the two-dimensional vectors here,

• Start by identifying corresponding components. For \(\mathbf{u}=2 \mathbf{i}-2 \mathbf{j}\), \(i\)-component is 2 and \(j\)-component is -2.
• For \(\mathbf{v}=-\mathbf{i}-\mathbf{j}\), \(i\)-component is -1 and \(j\)-component is -1.

Calculating the dot product involves:

• Multiplying the \(i\)-components: \(2 \times (-1) = -2\)
• Multiplying the \(j\)-components: \((-2) \times (-1) = 2\)
• Adding the products: \(-2 + 2 = 0\)

When the dot product is zero, vectors are orthogonal, as they meet at a right angle, having no influence on each other's direction.

Vector Components

Vectors are often expressed using their components, which decompose a vector into its basic parts. These parts include direction indicators, commonly represented by unit vectors like \(\mathbf{i}\) and \(\mathbf{j}\) in two dimensions.

• For example, in \(\mathbf{u} = 2 \mathbf{i} - 2 \mathbf{j}\), the component along the \(x\)-axis is 2 (attached to \(\mathbf{i}\)) and the component along the \(y\)-axis is -2 (attached to \(\mathbf{j}\)).
• Each vector component can give insight into the vector’s position and direction within a plane.

Understanding vector components can assist in breaking down complex mathematical problems into smaller, more manageable parts, aiding in operations like the dot product.

Vector Mathematics

Vector mathematics is a branch of mathematics dealing with vectors, which are quantities having both magnitude and direction. In analytical geometry, vectors are used to describe physical quantities like force, velocity, and acceleration.

• A vector in the Cartesian plane is often described using unit vectors \(\mathbf{i}\) and \(\mathbf{j}\), which ensure dimensional consistency.
• Operations like dot product and cross product allow us to examine interactions between vectors more profoundly.
• In problems where finding angles or determining orthogonality is necessary, vectors become invaluable tools.

Grasping vector mathematics opens doors to solving a wide array of physical problems, providing critical insights into how different elements interact in space.