Problem 50 - Algebra and Trigonometry

Question

Determine whether \(\mathbf{v}\) and \(\mathbf{w}\) are parallel, orthogonal, or neither. $$ \mathbf{v}=-2 \mathbf{i}+3 \mathbf{j}, \quad \mathbf{w}=-6 \mathbf{i}-4 \mathbf{j} $$

Step-by-Step Solution

Answer

The vectors \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal.

Step 1: Write the vectors

The vectors are written as, \(\mathbf{v}=-2 \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{w}=-6 \mathbf{i}-4\mathbf{j}\).

Step 2: Calculate Dot Product

The dot product of two vectors is obtained by multiplying corresponding components and adding those products. Therefore, \(\mathbf{v} \centerdot \mathbf{w} =(-2*-6) +(3*-4) =12 -12 =0.\)

Step 3: Determine if Vectors are Parallel

Vectors are parallel if one is a scalar multiple of the other. Comparing the ratios of the corresponding components of \(\mathbf{v}\) and \(\mathbf{w}\), we get, \(\frac{-2}{-6} \neq \frac{3}{-4}\). So, \(\mathbf{v}\) and \(\mathbf{w}\) aren't parallel.

Step 4: Conclude the relationship between \(\mathbf{v}\) and \(\mathbf{w}\)

Since the dot product of \(\mathbf{v}\) and \(\mathbf{w}\) equals 0 and \(\mathbf{v}\) is not a scalar multiple of \(\mathbf{w}\), \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal (perpendicular) to each other.

Key Concepts

Dot Product

The dot product is a fundamental concept in vector algebra. It is a way of multiplying two vectors to obtain a scalar value. This operation helps determine how two vectors relate in terms of their direction and angle towards each other.
To calculate the dot product of two vectors, follow these simple steps:

• Multiply the corresponding components of each vector together.
• Sum these products.

For example, if you have vectors \(\mathbf{v} = a\mathbf{i} + b\mathbf{j}\) and \(\mathbf{w} = c\mathbf{i} + d\mathbf{j}\), the dot product \(\mathbf{v} \cdot \mathbf{w}\) will be \(ac + bd\). In our exercise, the vectors \(\mathbf{v} = -2\mathbf{i} + 3\mathbf{j}\) and \(\mathbf{w} = -6\mathbf{i} - 4\mathbf{j}\) yield a dot product of 0:
\((-2)(-6) + (3)(-4) = 12 - 12 = 0\). This result is crucial as it leads us to the next topic of orthogonal vectors.

Orthogonal Vectors

Orthogonal vectors are two vectors that meet at a right angle, making them perpendicular to each other in a geometric sense. In vector algebra, this orthogonality is expressed when the dot product of the two vectors equals zero.
This is a unique feature because it highlights that there is no 'shared' direction between the two vectors. Instead, they spread out in completely different directions.In our exercise, since the dot product \(\mathbf{v} \cdot \mathbf{w} = 0\), vectors \(\mathbf{v}\) and \(\mathbf{w}\) are orthogonal. It's important to note that the zero result indicates that these vectors do not "work together" in a particular direction, emphasizing their perpendicular nature. By understanding this relationship better, students can tackle a variety of vector-related problems with greater ease.

Scalar Multiple

A scalar multiple involves scaling a vector by multiplying it by a scalar (a real number). This operation stretches or shrinks the vector but does not change its direction if the scalar is positive. If negative, it reverses the vector direction.
Determining whether two vectors are parallel is closely tied to the concept of scalar multiples. Two vectors can be considered parallel if one vector is a scalar multiple of the other.In our exercise, checking for parallelism involves comparing the ratios of corresponding components of vectors \(\mathbf{v}\) and \(\mathbf{w}\). For \(\mathbf{v} = -2\mathbf{i} + 3\mathbf{j}\) and \(\mathbf{w} = -6\mathbf{i} - 4\mathbf{j}\), the ratios are \(\frac{-2}{-6}\) and \(\frac{3}{-4}\). Since these ratios are not equal, \(\mathbf{v}\) and \(\mathbf{w}\) are not scalar multiples, hence not parallel.
Understanding and using scalar multiples can greatly accelerate learning about vector interrelations, whether addressing more complex vector transformations or solving everyday problems.