Problem 27 - Precalculus : Building Concepts and Connections

Question

Determine whether the given pairs of vectors are orthogonal. $$\mathbf{v}=\langle 2,-3\rangle, \mathbf{w}=\langle-6,4\rangle$$

Step-by-Step Solution

Answer

The vectors \( \mathbf{v}=\langle 2,-3\rangle, \mathbf{w}=\langle-6,4\rangle \) are not orthogonal because their dot product is not equal to 0.

Step 1: Calculate the dot product

The dot product of two vectors \( \mathbf{v}=\langle v_1, v_2 \rangle \) and \( \mathbf{w}=\langle w_1, w_2 \rangle \) in 2-dimensional space is calculated as \( \mathbf{v} \cdot \mathbf{w} = v_1*w_1+v_2*w_2 \). Following this, the dot product of vectors \( \mathbf{v}=\langle 2,-3 \rangle \) and \( \mathbf{w}=\langle -6,4 \rangle \) is \( \mathbf{v} \cdot \mathbf{w} = 2*(-6) +(-3)*4 \).

Step 2: Evaluate the dot product

We substitute the values and calculate to find \( \mathbf{v} \cdot \mathbf{w} = -12 -12 = -24 \).

Step 3: Determine Orthogonality

Vectors are orthogonal if their dot product equals 0. Since \( \mathbf{v} \cdot \mathbf{w} = -24 \), which is not 0, the vectors are not orthogonal.

Key Concepts

Dot Product

The dot product is a mathematic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is crucial in vector algebra, especially when dealing with perpendicularity or projection.

The computational formula for the dot product in 2-dimensional space for two vectors, say \( \mathbf{v} \), and \( \mathbf{w} \) with components \( \mathbf{v}=\langle v_1, v_2 \rangle \) and \( \mathbf{w}=\langle w_1, w_2 \rangle \) respectively, is:\
\[ \mathbf{v} \cdot \mathbf{w} = v_1 w_1 + v_2 w_2 \]

This scalar result can indicate the magnitude of projection of one vector onto another when the vectors are not orthogonal. Moreover, the sign of the dot product can show the general direction of one vector with respect to another.

Vector Orthogonality

Vector orthogonality is the property saying that two vectors are perpendicular to each other in space. Mathematically, two vectors are orthogonal if their dot product is zero. This is because a zero dot product implies that there is no projection of one vector onto another.

In the context of the exercise provided, after calculating the dot product and finding a non-zero result, it's clearly established that vectors \( \mathbf{v} \) and \( \mathbf{w} \) are not orthogonal. To improve understanding, visualize vector orthogonality by thinking of the x and y axes on a two-dimensional graph. The axes are orthogonal to each other as they intersect at a right angle and their dot product equals zero.

2-Dimensional Vectors

Vectors in two-dimensional space, also known as 2D vectors, are defined by two components that determine their direction and magnitude. These components can be represented as ordered pairs \( \langle x, y \rangle \) which correspond to their position on the Cartesian plane.

Visual Representation

In diagrams, these vectors often appear as arrows pointing from the origin (0,0) to a point defined by the x and y-coordinates. The x-component affects the vector's horizontal position, while the y-component controls the vertical position. Understanding 2D vectors is fundamental when learning about vector operations, such as addition, subtraction, and scalar multiplication, along with the dot product and orthogonality.