Problem 18 - APEX Calculus

Question

A vector \(\vec{v}\) is given. Give two vectors that are orthogonal to \(\vec{v}\). \(\vec{v}=\langle-3,5\rangle\)

Step-by-Step Solution

Answer

Two orthogonal vectors to \(\vec{v}\) are \(\langle 5, 3 \rangle\) and \(\langle -5, -3 \rangle\).

Step 1: Understand the Problem

We need to find two vectors that are orthogonal (perpendicular) to a given vector \(\vec{v} = \langle -3, 5 \rangle\). In a 2D plane, a vector \(\langle a, b \rangle\) is orthogonal to \(\langle b, -a \rangle\) or its negative \(\langle -b, a \rangle\).

Step 2: Identify Orthogonal Vectors

For a vector \(\vec{v} = \langle -3, 5 \rangle\), an orthogonal vector can be found by swapping its components and changing one sign. This gives us the vector \(\langle 5, 3 \rangle\). Another orthogonal vector by changing the other sign is \(\langle -5, -3 \rangle\).

Step 3: Verify Orthogonality

Two vectors \(\langle a_1, b_1 \rangle\) and \(\langle a_2, b_2 \rangle\) are orthogonal if their dot product is zero: \(a_1 \cdot a_2 + b_1 \cdot b_2 = 0\). For our vectors: \(-3 \times 5 + 5 \times 3 = 0\) and \(-3 \times -5 + 5 \times -3 = 0\). Both give 0, confirming orthogonality.

Key Concepts

Dot Product

The dot product is a crucial concept when dealing with vectors. It helps determine whether two vectors are orthogonal, or perpendicular, to each other. To calculate the dot product of two vectors \(\vec{a} = \langle a_1, b_1 \rangle\)and \(\vec{b} = \langle a_2, b_2 \rangle\)in 2D space, we perform the operation: \(a_1 \cdot a_2 + b_1 \cdot b_2\).

• If the dot product is zero, it means the vectors are orthogonal.
• The calculation involves individual multiplication of corresponding components followed by their sum.

For example, considering our vectors \(\vec{v} = \langle -3, 5 \rangle\)and \(\vec{w} = \langle 5, 3 \rangle\), the dot product is calculated as: \(-3 \times 5 + 5 \times 3 = 0\).The result is zero, so these vectors are indeed orthogonal. The dot product is a straightforward way to check orthogonality.

2D Vectors

In a 2D plane, vectors are represented by two components which show direction and magnitude. A vector \(\vec{v} = \langle a, b \rangle\)is essentially a directed line segment from the origin. It has two main properties:

• The direction, typically determined by the angle with respect to a reference axis.
• The magnitude, which is the length of the vector calculated as \(\sqrt{a^2 + b^2}\).

Vectors are widely used to represent quantities like force and velocity in physics. When we discuss vectors being orthogonal, it means they have no component in the same direction, making them perfectly perpendicular.

Understanding vectors in 2D is fundamental, as many real-world phenomena find applications with these simple representations.

Vector Components

Vector components describe the individual contributions of each part of a vector in their respective orthogonal directions. In a 2D vector like \(\vec{v} = \langle -3, 5 \rangle\), the components \(a = -3\) and \(b = 5\) represent its projection along the x and y axes, respectively.

• Each component indicates how much the vector moves in a specific direction.
• When working with orthogonal vectors, swapping components with one sign change creates a new vector that is perpendicular.

For instance, from \(\vec{v} = \langle -3, 5 \rangle\), the orthogonal vector \(\langle 5, 3 \rangle\)is derived by swapping components and changing one sign. This transformation is key in finding vectors that are perpendicular in the plane.

Considering vector components helps visualize and manipulate vectors, offering insight into their behavior and interaction, especially in identifying orthogonality.