Problem 13 - 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. $$ \mathbf{a}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{b}=-\frac{12}{7} \mathbf{i}+\frac{20}{7} \mathbf{j} $$

Step-by-Step Solution

Answer

The vectors are parallel and have opposite directions.

Step 1: Check for Parallelism

Two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel if one is a scalar multiple of the other, i.e., \( \mathbf{b} = k \mathbf{a} \) for some scalar \( k \). First, check if \( -\frac{12}{7} = 3k \) and \( \frac{20}{7} = -5k \). This can be formed into equations:\[3k = -\frac{12}{7} \quad \Rightarrow \quad k = -\frac{4}{7}\]\[-5k = \frac{20}{7} \quad \Rightarrow \quad k = -\frac{4}{7}\]Both \( k \) values are equal, confirming that \( \mathbf{a} \) and \( \mathbf{b} \) are parallel.

Step 2: Determine the Direction

Calculate the value of \( k \) and its sign to determine the direction. Since both components provide \( k = -\frac{4}{7} \), which is negative, the vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel and have opposite directions.

Key Concepts

Scalar Multiple

To understand if two vectors are parallel, we need to check if they are scalar multiples of one another. This means that you can express one vector as a number (scalar) times the other vector. Mathematically, if you have vectors \( \mathbf{a} \) and \( \mathbf{b} \), they are parallel if \( \mathbf{b} = k \mathbf{a} \) where \( k \) is a scalar. In our example, vector \( \mathbf{a} = 3\mathbf{i} - 5\mathbf{j} \) and vector \( \mathbf{b} = -\frac{12}{7}\mathbf{i} + \frac{20}{7}\mathbf{j} \). To find \( k \), compare the coefficients of corresponding components:

• For \( \mathbf{i} \), set \( 3k = -\frac{12}{7} \).
• For \( \mathbf{j} \), set \( -5k = \frac{20}{7} \).

Solve these equations:

• \( 3k = -\frac{12}{7} \) gives \( k = -\frac{4}{7} \).
• \( -5k = \frac{20}{7} \) also gives \( k = -\frac{4}{7} \).

Both calculations produce the same scalar \( k \), confirming that the vectors are indeed scalar multiples and thus parallel.

Vector Direction

The direction of a vector is simply the path that the vector points. In terms of parallel vectors, they can either have the same direction or opposite directions. When two vectors are parallel, analyzing the scalar multiple \( k \) gives you insight into their direction:

• If \( k > 0 \), the vectors have the same direction.
• If \( k < 0 \), the vectors point in opposite directions.

Using our example, since \( k = -\frac{4}{7} \) is negative, it indicates that the two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), do not point the same way. Instead, they follow opposite paths even though they are aligned along the same line, establishing their parallel nature but with different heading directions.

Opposite Directions

Parallel vectors can point in opposite directions if one is a negative scalar multiple of the other. This means they lie on the same line but point in reverse directions.With vectors \( \mathbf{a} \) and \( \mathbf{b} \), since both resulted in the negative scalar \( k = -\frac{4}{7} \), it confirms that vector \( \mathbf{b} \) is \( -\frac{4}{7} \) times vector \( \mathbf{a} \), enforcing the idea they move in opposite directions.Understanding opposite directions helps you visualize scenarios like two forces canceling each other out if applied in parallel opposition, or objects moving away from each other on the same path. This knowledge is crucial in physics and engineering where force direction impacts momentum and motion dynamics.