Problem 29 - Algebra and Trigonometry

Question

Use the dot product to determine whether v and w are orthogonal. $$ \mathbf{v}=3 \mathbf{i}, \quad \mathbf{w}=-4 \mathbf{i} $$

Step-by-Step Solution

Answer

The vectors v and w are not orthogonal.

Step 1: Write out the vectors

Initially print out the vectors. The vector \(\mathbf{v}\) is \(3 \mathbf{i}\), and \(\mathbf{w}\) is \(-4 \mathbf{i}\)

Step 2: Compute the dot product

Compute the dot product of \(\mathbf{v}\) and \(\mathbf{w}\). The dot product of two vectors, \(\mathbf{v}\cdot\mathbf{w}\), is calculated by multiplying the corresponding components of each vector and then summing those products. In this case, \(\mathbf{v}\cdot\mathbf{w}\) is \(3*-4 = -12\)

Step 3: Check orthogonality

If the dot product is zero, the two vectors are orthogonal (perpendicular). In this case, since the dot product of v and w equals -12, v and w are not orthogonal to each other.

Key Concepts

Understanding Vectors

Vectors are mathematical entities that have both magnitude and direction. They are often represented by arrows in diagrams or by unit vectors such as \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) when working in three-dimensional space. In our example, \(\mathbf{v}=3 \mathbf{i}\) and \(\mathbf{w}=-4 \mathbf{i}\) are vectors in one dimension, where \(\mathbf{i}\) can be understood as a representation of the unit vector along the x-axis.

Understanding vectors involves:

• Magnitude: This is the length or size of the vector.
• Direction: This dictates the vector's orientation.
• Components: These are the projections of the vector along the coordinate axes, determined by multiplying each basis unit vector (like \(\mathbf{i}\), \(\mathbf{j}\)) by a scalar coefficient.

By learning to manipulate these characteristics, you can solve many problems involving forces, velocities, and other vector-related phenomena.

Exploring Orthogonality

Orthogonality is a concept within vector mathematics where two vectors are perpendicular to each other. This occurs when the dot product of the two vectors equals zero. It's similar to saying that at a right angle, vectors do not influence each other along their lines of action.

For instance, in our exercise, we wanted to check if \(\mathbf{v}\) and \(\mathbf{w}\) were orthogonal. To do this, we calculated their dot product: \[ \mathbf{v}\cdot\mathbf{w} = 3 \cdot (-4) = -12 \] Since the result is not zero, \(\mathbf{v}\) and \(\mathbf{w}\) are not orthogonal.

Orthogonality is important in many fields, such as physics and engineering, as it often implies independent directions or separation of concerns, where certain component actions do not interfere with others.

The Dot Product and Component-wise Multiplication

The dot product is a mathematical operation that combines two vectors, summarizing their mutual influence along the same direction. It's calculated by multiplying corresponding components of the vectors and then summing the results. This operation is central to understanding many vector relationships, like determining orthogonality and projections.

Component-wise multiplication refers to the step within the dot product calculation where each part of the vectors is multiplied individually.For vectors \(\mathbf{v}\) and \(\mathbf{w}\):

• The vector components \(3\) and \(-4\) are multiplied to yield \(-12\).

The resulting scalar tells us whether the vectors share the same direction (positive dot product), are perpendicular (dot product zero), or point in opposite directions (negative dot product). In this case, their negative dot product tells us that \(\mathbf{v}\) and \(\mathbf{w}\) partially cancel each other out sign-wise rather than being perpendicular.