Problem 15 - 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. $$ \mathbf{a}=\left\langle\frac{2}{3}, \frac{1}{2}\right\rangle, \quad \mathbf{b}=\langle 8,6\rangle $$

Step-by-Step Solution

Answer

The vectors are parallel and have the same direction.

Step 1: Check Proportionality of Components

To determine if two vectors are parallel, their components must be proportional. This means there must exist a scalar \(k\) such that \(\frac{2}{3} = k \times 8\) and \(\frac{1}{2} = k \times 6\).

Step 2: Solve for the Scalar k

To find the scalar \(k\), set up the equation for one pair of components. Using the first pair, \(\frac{2}{3} = k \times 8\). Solve for \(k\): \[ k = \frac{2}{3} \div 8 = \frac{2}{3} \times \frac{1}{8} = \frac{1}{12} \].

Step 3: Verify Scalar with Second Component Pair

Confirm that the same \(k\) works for the second component pair. Solve \(\frac{1}{2} = \frac{1}{12} \times 6\). Calculating, \(\frac{1}{12} \times 6 = \frac{6}{12} = \frac{1}{2}\), which matches the second component of \( \mathbf{a} \). Thus, the proportionality is verified for both component pairs.

Step 4: Determine Direction of Vectors

Since both proportionalities exist using the same positive scalar \(k = \frac{1}{12}\), the vectors have the same direction. If \(k\) were negative, they would have opposite directions.

Key Concepts

Vector Proportionality

When two or more vectors are said to be parallel, it implies there is some form of proportionality between their components. This means the vectors can be expressed as being multiplied by a scalar quantity that relates them. For example, if we have vectors \( \mathbf{a} = \langle x_1, y_1 \rangle \) and \( \mathbf{b} = \langle x_2, y_2 \rangle \), they are parallel if there exists a scalar \( k \) such that:

• \( x_1 = k \cdot x_2 \)
• \( y_1 = k \cdot y_2 \)

This concept is what we call vector proportionality. The same scalar \( k \) must apply to both components of the vector. If a suitable \( k \) doesn't exist, the vectors are not parallel. Ensuring that both parts of the vector equations satisfy this is crucial as it confirms the vectors run parallel along the same gradient or slope.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar value — a single number. This operation stretches or shrinks the vector without changing its direction. If a vector \( \mathbf{v} = \langle v_1, v_2 \rangle \), and it's multiplied by a scalar \( k \), the resulting vector is \( k \mathbf{v} = \langle k \cdot v_1, k \cdot v_2 \rangle \).
The whole process affects the magnitude but not the direction unless the scalar is negative. A negative scalar will flip the vector's direction. Thus, scalar multiplication is essential to understanding vector parallelism because it helps establish the constant \( k \) that confirms whether one vector is just a magnified or shrunken version of another.

Vector Direction

The direction of a vector is essentially the angle at which the vector points in a two-dimensional or three-dimensional space. For vectors to have the same direction, the scalar used in scalar multiplication should be positive. This means the vectors will lie on the same line or path and point in the same direction along that path.
For example, if vectors \( \mathbf{a} \) and \( \mathbf{b} \) have been established as parallel with a positive scalar \( k \), they share the same direction. If, on the other hand, the scalar is negative, then the vectors point in opposite directions, although they remain parallel.
Direction is integral to understanding vector relationships because, even if vectors are parallel, knowing the direction helps determine how they relate spatially — whether they cooperate (same direction) or counteract (opposite direction). This directional insight gives a complete picture of how vectors interact in space.