Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 9 -12: Show that the vectors are orthogonal. $$ -4 \mathbf{j}, \quad-7 \mathbf{i} $$

Step-by-Step Solution

Answer

The vectors are orthogonal because their dot product is zero.

Step 1: Identify the Vectors

The vectors given in this problem are \[ \mathbf{a} = -4 \mathbf{j} = \begin{bmatrix} 0 \ -4 \end{bmatrix} \] and \[ \mathbf{b} = -7 \mathbf{i} = \begin{bmatrix} -7 \ 0 \end{bmatrix} \]. These are the component forms of the vectors in the Cartesian plane.

Step 2: Understand Orthogonality

Two vectors are orthogonal if their dot product is zero. The dot product of vectors \(\mathbf{a}\) and \(\mathbf{b}\) is calculated using the formula:\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \] where \( a_1, a_2 \) are components of \( \mathbf{a} \) and \( b_1, b_2 \) are components of \( \mathbf{b} \).

Step 3: Calculate the Dot Product

For the vectors \( \mathbf{a} = \begin{bmatrix} 0 \ -4 \end{bmatrix} \) and \( \mathbf{b} = \begin{bmatrix} -7 \ 0 \end{bmatrix} \), compute their dot product:\[\mathbf{a} \cdot \mathbf{b} = (0)(-7) + (-4)(0) = 0 + 0 = 0\] Since the dot product is 0, the vectors are orthogonal.

Key Concepts

Dot Product

The dot product is a way to multiply two vectors that results in a scalar. Imagine it like a mathematical tool to measure how much one vector extends in the direction of another. The formula to calculate the dot product of two vectors, say \( \mathbf{a} \) and \( \mathbf{b} \), is as follows:\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \]where:

• \( a_1 \) and \( a_2 \) are the components of \( \mathbf{a} \)
• \( b_1 \) and \( b_2 \) are the components of \( \mathbf{b} \)

This operation gives insight into the angle between the vectors. If the dot product equals zero, the vectors are orthogonal, meaning they meet at a 90-degree angle.In our exercise, calculating the dot product of \( \mathbf{a} = \begin{bmatrix} 0 \ -4 \end{bmatrix} \) and \( \mathbf{b} = \begin{bmatrix} -7 \ 0 \end{bmatrix} \) reveals it is zero. Therefore, confirming their orthogonality.

Vector Components

Vector components are the "building blocks" of a vector, often described by how far the vector goes along each axis of a plane. In a two-dimensional space, a vector \( \mathbf{v} \) can be broken down into its components as follows:\[ \mathbf{v} = \begin{bmatrix} v_1 \ v_2 \end{bmatrix} \]where:

• \( v_1 \) is the component along the x-axis (i-direction)
• \( v_2 \) is the component along the y-axis (j-direction)

Looking at our vectors:

• \( \mathbf{a} = \begin{bmatrix} 0 \ -4 \end{bmatrix} \) is purely along the y-axis with no x-component.
• \( \mathbf{b} = \begin{bmatrix} -7 \ 0 \end{bmatrix} \) lies entirely along the x-axis, having no y-component.

Understanding vector components is crucial because it makes calculations, such as that of the dot product, straightforward. Each separate component contributes its part in these operations.

Cartesian Plane

The Cartesian plane is a coordinate system that uses two perpendicular axes to define a plane where vectors can be plotted. It consists of an x-axis (horizontal) and a y-axis (vertical). Each point on this plane is represented by a set of coordinates \( (x, y) \), where:

• \( x \) is the horizontal position
• \( y \) is the vertical position

Vectors in the Cartesian plane are often expressed in the form of component vectors, such as our \( \mathbf{a} \) and \( \mathbf{b} \). This makes it easier to perform vector operations and visualize the vectors' positions.For example, vector \( \mathbf{a} = -4 \mathbf{j} \) represents a point 4 units down along the y-axis. Meanwhile, vector \( \mathbf{b} = -7 \mathbf{i} \) signifies a point 7 units to the left along the x-axis. Knowing this helps us conclude why their dot product is zero—they're perpendicular, occupying exclusive directions on the Cartesian plane.