Problem 45 - Algebra and Trigonometry

Question

Determine whether \(\mathbf{v}\) and \(\mathbf{w}\) are parallel, orthogonal, or neither. $$ \mathbf{v}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}-10 \mathbf{j} $$

Step-by-Step Solution

Answer

The vectors \(\mathbf{v} = 3\mathbf{i} - 5\mathbf{j}\) and \(\mathbf{w} = 6\mathbf{i} - 10\mathbf{j}\) are parallel to each other.

Step 1: Identify the Vectors

The given vectors are \(\mathbf{v} = 3\mathbf{i} - 5\mathbf{j}\) and \(\mathbf{w} = 6\mathbf{i} - 10\mathbf{j}\).

Step 2: Check for Proportionality

Vectors \(\mathbf{v}\) and \(\mathbf{w}\) are proportional if format \(a\mathbf{v} = \mathbf{w}\), where \(a\) is a scalar other than zero, holds. Divide vector \(\mathbf{w}\) by vector \(\mathbf{v}\), term-by-term to see if the ratio across the i and j terms are the same. After simplification, the ratio is 2 which is constant across all terms, so this implies \(\mathbf{v}\) and \(\mathbf{w}\) are proportional.

Step 3: Conclusion

As \(\mathbf{v}\) and \(\mathbf{w}\) are proportional, the vectors are parallel. There is no need to check for orthogonality since parallel vectors are not orthogonal.

Key Concepts

Vectors

Vectors are fundamental in mathematics, representing quantities that have both magnitude and direction. They come in various components, such as the

• i component, which denotes the horizontal direction
• j component, which denotes the vertical direction

These components work together to define the vector's position in a plane. In our original problem, we have two vectors:

• \( \mathbf{v} = 3\mathbf{i} - 5\mathbf{j} \)
• \( \mathbf{w} = 6\mathbf{i} - 10\mathbf{j} \)

Understanding vectors requires visualizing them as arrows. The tail of each arrow is usually at the origin, and the head points to the position described by the vector elements.

Proportionality

Proportionality between vectors means that one vector can be expressed as a scalar multiple of another. For vectors, the relationship becomes clear when you compare their components:
If every component of vector \( \mathbf{v} \) can be obtained by multiplying the corresponding component of another vector \( \mathbf{w} \) by a constant scalar, then \( \mathbf{v} \) is proportional to \( \mathbf{w} \).
In our problem, vector \( \mathbf{w} \) can be seen as twice vector \( \mathbf{v} \), since:

• \( 6\mathbf{i} = 2 \times 3\mathbf{i} \)
• \( -10\mathbf{j} = 2 \times -5\mathbf{j} \)

This constant scalar (2 in this case) shows that the vectors are proportional, indicating a specific alignment or parallelism.

Orthogonal Vectors

Orthogonal vectors are vectors that are at right angles to one another. This orthogonality is true if the dot product of the vectors is zero. The dot product is calculated by multiplying corresponding components and adding the results:
For two vectors, \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \), the dot product is:\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2\]
In cases where the vectors are parallel, they are not orthogonal. Our original problem showed that because \( \mathbf{v} \) and \( \mathbf{w} \) are proportional, they cannot be orthogonal. You can imagine orthogonal vectors as making a 90-degree angle, like the axes of an "L" shape.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a single number). This operation will change the magnitude of the vector but not its direction.
The scalar multiplication of a vector \( \mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} \) by a scalar "a" results in:\[a\mathbf{v} = a(v_1\mathbf{i}) + a(v_2\mathbf{j}) = (av_1)\mathbf{i} + (av_2)\mathbf{j}\]
In our example, vector \( \mathbf{w} = 6\mathbf{i} - 10\mathbf{j} \) can be seen as the product of vector \( \mathbf{v} = 3\mathbf{i} - 5\mathbf{j} \) and the scalar 2:

• \( 6\mathbf{i} = 2 \times 3\mathbf{i} \)
• \( -10\mathbf{j} = 2 \times -5\mathbf{j} \)

Scalar multiplication is a simple and effective way to see how vectors relate in magnitude while keeping their direction consistent.