Problem 64 - College Trigonometry

Question

We say that two non-zero vectors \(\vec{v}\) and \(\vec{w}\) are parallel if they have same or opposite directions. That is, \(\vec{v} \neq \overrightarrow{0}\) and \(\vec{w} \neq \overrightarrow{0}\) are parallel if either \(\hat{v}=\hat{w}\) or \(\hat{v}=-\hat{w}\). Show that this means \(\vec{v}=k \vec{w}\) for some non- zero scalar \(k\) and that \(k>0\) if the vectors have the same direction and \(k<0\) if they point in opposite directions.

Step-by-Step Solution

Answer

If \(\vec{v}\) and \(\vec{w}\) are parallel, \(\vec{v}=k\vec{w}\) for some non-zero scalar \(k\), where \(k>0\) if directions are same and \(k<0\) if directions are opposite.

Step 1: Understanding Vector Directions

Two non-zero vectors \(\vec{v}\) and \(\vec{w}\) are parallel if they either have the same direction (\(\hat{v}=\hat{w}\)) or opposite directions (\(\hat{v}=-\hat{w}\)). Unit vectors \(\hat{v}\) and \(\hat{w}\) indicate the direction of vectors. Thus, if \(\vec{v}\) and \(\vec{w}\) are parallel, their relationship can express the magnitude and direction.

Step 2: Express Vectors in Term of Scalars

Considering the relations \(\hat{v}=\hat{w}\) and \(\hat{v}=-\hat{w}\), this implies that there exists some scalar \(k\) such that \(\vec{v} = k\vec{w}\). The vector \(\vec{v}\) is expressed as a scalar multiple of \(\vec{w}\), where \(k\) determines both the magnitude and direction relationship between \(\vec{v}\) and \(\vec{w}\).

Step 3: Determine the Sign of the Scalar

If \(\hat{v}=\hat{w}\), then \(\vec{v}\) and \(\vec{w}\) are in the same direction, so \(k\) is positive (i.e., \(k>0\)). Conversely, if \(\hat{v}=-\hat{w}\), then \(\vec{v}\) and \(\vec{w}\) are in opposite directions, causing \(k\) to be negative (i.e., \(k<0\)). This division helps in understanding that the scalar \(k\) also indicates direction along with proportionality.

Key Concepts

Scalar Multiplication

When we talk about scalar multiplication in the context of vectors, we mean scaling a vector by a certain factor. This factor is known as a scalar and is typically a real number. If you have a vector \(\vec{w}\), multiplying it by a scalar \(k\) results in a new vector \(\vec{v} = k\vec{w}\). - **Scaling Up and Down**: The scalar dictates whether we scale the vector up or down.- **Changing Direction**: If the scalar is negative, it not only scales the vector but also reverses its direction.Thus, scalar multiplication helps in modifying both the size and sometimes the direction of a vector, depending on the value of the scalar. For example, a scalar of 2 doubles the vector’s magnitude, while a scalar of -2 doubles the magnitude and reverses the direction.

Vector Direction

The direction of a vector is essential for understanding how vectors relate to each other, especially when determining if they are parallel. Every vector has both a magnitude (i.e. size) and a direction.- **Unit Vectors as Direction Indicators**: The unit vector, denoted by \(\hat{v}\) or \(\hat{w}\), captures the direction component of a vector but with a magnitude of 1.- **Parallel Vectors**: Two vectors \(\vec{v}\) and \(\vec{w}\) are parallel if their direction vectors \(\hat{v}\) and \(\hat{w}\) are either identical (\(\hat{v} = \hat{w}\)) or opposite (\(\hat{v} = -\hat{w}\)).Understanding vector direction allows us to assess the alignment between vectors, which is crucial in numerous mathematical applications, such as physics, where understanding forces and directions is necessary.

Unit Vectors

Unit vectors are special vectors that have a magnitude of 1. They are primarily used to indicate direction and make calculations involving direction independent of magnitude.- **Normalization**: Any vector \(\vec{v}\) can be made into a unit vector by dividing it by its magnitude \(|\vec{v}|\). The resulting unit vector \(\hat{v}\) maintains the original vector's direction.- **Significance in Parallel Vectors**: In parallel vector analysis, unit vectors help simplify understanding by focusing solely on direction. Therefore, when two vectors are parallel, their corresponding unit vectors either match or are precise opposites.Unit vectors simplify many problems involving direction, serving as a foundational tool in areas like computer graphics, where directions without scale are crucial.