Problem 4 - Algebra and Trigonometry

Question

If \(\mathbf{v}=3 \mathbf{w},\) then the two vectors \(\mathbf{v}\) and \(\mathbf{w}\) are _________.

Step-by-Step Solution

Answer

The two vectors are parallel.

Step 1: Understand the Relationship Between Vectors

The problem states that \(\backslashmathbf\brace{v}=3\backslashmathbf\brace{w}\). This means that vector \(\backslashmathbf\brace{v}\) is three times the vector \(\backslashmathbf\brace{w}\). To put it another way, \(\backslashmathbf\brace{v}\) is obtained by scaling \(\backslashmathbf\brace{w}\) by a factor of 3.

Step 2: Identify Vector Parallelism

Two vectors are parallel if one is a scalar multiple of the other. Since \(\backslashmathbf\brace{v}=3\backslashmathbf\brace{w}\), \(\backslashmathbf\brace{v}\) is a scalar multiple of \(\backslashmathbf\brace{w}\). Therefore, \(\backslashmathbf\brace{v}\) and \(\backslashmathbf\brace{w}\) are parallel vectors.

Key Concepts

Scalar Multiplication

In the given problem, we see that \(\backslashmathbf\brace{v}=3\backslashmathbf\brace{w}\). This tells us that vector \(\backslashmathbf\brace{v}\) is three times the vector \(\backslashmathbf\brace{w}\). To grasp this, consider what happens when we multiply any vector by a scalar (a real number).

Scalar multiplication involves multiplying each component of a vector by the same scalar. For example, if \(\backslashmathbf\brace{w} = (x, y, z)\) and \ k = 3 \,then multiplying vector \(\backslashmathbf\brace{w}\) by scalar 3 gives us \(\backslashmathbf\brace{v} = (3x, 3y, 3z)\). This shows how each component of the original vector \(\backslashmathbf\brace{w}\) is adjusted by the scalar multiplicative factor (3 in this case).

Key points to remember about scalar multiplication:

• Each component of the vector gets multiplied by the scalar.
• The direction of the vector stays the same if the scalar is positive, but the magnitude (length) changes.
• If the scalar is negative, the direction of the vector reverses.

Vector Relationships

Understanding the relationship between vectors is critical in vector mathematics. Here, we are dealing with two vectors \(\backslashmathbf\brace{v}\) and \(\backslashmathbf\brace{w}\), and we need to find out how they relate to one another given that \(\backslashmathbf\brace{v}=3\backslashmathbf\brace{w}\).

Several properties define vector relationships:

• Two vectors are equal if they have the same magnitude and direction.
• Vectors can be added or subtracted component-wise to form new vectors.
• Multiplying by a scalar changes the magnitude but not the direction (unless the scalar is negative, which reverses the direction).

In this specific problem, the scalar multiplication relation indicates that vectors \(\backslashmathbf\brace{v}\) and \(\backslashmathbf\brace{w}\) don't just correlate but also share a direct proportional relationship. Hence, \(\backslashmathbf\brace{v}\) is a scaled version of \(\backslashmathbf\brace{w}\), confirming a specific type of vector relationship called parallelism.

Vector Parallelism

Parallel vectors share a unique relationship where one vector can be expressed as a scalar multiple of the other. This is exactly what our problem illustrates with \(\backslashmathbf\brace{v}=3\backslashmathbf\brace{w}\).

To understand vector parallelism better, consider the following points:

• Two vectors \(\backslashmathbf\brace{a}\) and \(\backslashmathbf\brace{b}\) are parallel if there exists a scalar \ k \, such that \(\backslashmathbf\brace{a} = k\backslashmathbf\brace{b}\).
• Parallel vectors always have the same or opposite direction depending on the sign of the scalar.
• The magnitude of the scalar determines how much one vector is stretched or shrunk compared to the other.

In our case, \(\backslashmathbf\brace{v}\) is obtained by multiplying \(\backslashmathbf\brace{w}\) by a scalar 3, demonstrating that \(\backslashmathbf\brace{v}\) is parallel to \(\backslashmathbf\brace{w}\). Understanding this concept of scalar multiplication leading to parallelism is fundamental in vector algebra.