Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-16: Show that the vectors are parallel, and determine whether they have the same direction or opposite directions. $$ \mathbf{a}=\langle 6,18\rangle, \quad \mathbf{b}=\langle-4,-12\rangle $$

Step-by-Step Solution

Answer

The vectors are parallel with opposite directions.

Step 1: Recognize Parallel Vectors

Vectors are parallel if one is a scalar multiple of the other. This means there exists a constant \( k \) such that \( \mathbf{b} = k \mathbf{a} \). For vectors \( \mathbf{a} = \langle 6, 18 \rangle \) and \( \mathbf{b} = \langle -4, -12 \rangle \), we need to verify if \( \mathbf{b} = k \mathbf{a} \).

Step 2: Find the Scalar Multiple k

To find \( k \), compare the corresponding components of the vectors. For the first components, we have \(-4 = 6k\), which gives us \( k = -\frac{2}{3} \). Check this \( k \) against the second components: \(-12 = 18k\). This also gives \( k = -\frac{2}{3} \). Both components give the same \( k \), confirming the vectors are parallel.

Step 3: Analyze Direction

Since \( k = -\frac{2}{3} \) is negative, the vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel but have opposite directions.

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental concept in vector operations. It involves multiplying a vector by a scalar, which is simply a real number. When you multiply a vector by a scalar, every component of the vector gets multiplied by that scalar. This operation can stretch or shrink the vector, but it does not change its direction unless the scalar is negative.

Consider a vector \( \mathbf{v} = \langle x, y \rangle \). When we multiply \( \mathbf{v} \) by a scalar \( k \), we get a new vector \( k \mathbf{v} = \langle kx, ky \rangle \).

The scalar multiplication affects:

• the length (or magnitude) of the vector.
• the direction of the vector depending on whether \( k \) is positive or negative.

A positive scalar does not alter the direction of the vector, while a negative scalar reverses the direction.

In our case, \( \mathbf{a} = \langle 6, 18 \rangle \) and \( \mathbf{b} = \langle -4, -12 \rangle \) are related by the scalar \( k = -\frac{2}{3} \), indicating both a change in magnitude and a reversal in direction due to the negative value of \( k \).

Vector Direction

A vector's direction is an essential property, representing the orientation of the vector in the coordinate system. It is defined relative to the axes and is often indicated as an angle or as a ratio of its components.

Vectors \( \mathbf{a} = \langle x_1, y_1 \rangle \) and \( \mathbf{b} = \langle x_2, y_2 \rangle \) are parallel if one is a scalar multiple of the other. Essentially, they have the same or exactly opposite directions. For two vectors to share the same direction, the scalar must be positive.

Analyzing direction:

• If both vectors point in the same way, the multiplication factor \( k \) will be positive.
• If \( k \) is negative, one vector points in the complete opposite direction compared to the other.

In our example, the negative \( k = -\frac{2}{3} \) indicates that while \( \mathbf{b} \) is parallel to \( \mathbf{a} \), it is in the opposite direction, meaning they face away from each other.

Opposite Directions

When vectors have opposite directions, they point directly away from each other in a coordinate plane. This scenario is mathematically proven by observing the scalar \( k \, (<0) \) obtained during scalar multiplication.

If vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel, and the scaling factor \( k \) is negative, the vectors are parallel but face opposite directions.

Key observations for opposite directions:

• The scalar \( k \) must be negative.
• The ratio of the corresponding components will be constant.
• The vectors have the same line of action but go opposite ways.

In our exercise, since both components have the same negative scalar \( k = -\frac{2}{3} \), it confirms that while \( \mathbf{a} \) and \( \mathbf{b} \) are parallel, they travel directly opposite to one another, making their paths mirror reflections but in reverse.

Two-Dimensional Vectors

Two-dimensional vectors are common in mathematics, representing quantities with both magnitude and direction in a plane. Each vector is represented by an ordered pair, such as \( \mathbf{a} = \langle x, y \rangle \).

These vectors can lie along or be parallel to coordinate axes, or exist anywhere in the plane. The relationship between two-dimensional vectors encompasses concepts such as addition, subtraction, and especially parallelism, as seen in our exercise.

Understanding two-dimensional vectors includes:

• Each pair \( \langle x, y \rangle \) defines a position relative to the origin.
• Their relationships, including parallelism, can be analyzed through their components.
• Operations with two-dimensional vectors follow simple rules that utilize elementary algebra.

For example, with \( \mathbf{a} = \langle 6, 18 \rangle \) and \( \mathbf{b} = \langle -4, -12 \rangle \), we explore their parallel nature and directionality through component analysis and scalar multiplication, both central features in two-dimensional vector studies.