Problem 20 - Algebra and Trigonometry

Question

Determine whether the given vectors are perpendicular. $$ \mathbf{u}=4 \mathbf{i}, \quad \mathbf{v}=-\mathbf{i}+3 \mathbf{j} $$

Step-by-Step Solution

Answer

The vectors \( \mathbf{u} \) and \( \mathbf{v} \) are not perpendicular.

Step 1: Understand Vector Perpendicularity

Two vectors are perpendicular if their dot product is zero. For vectors \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{v} = c\mathbf{i} + d\mathbf{j} \), the dot product is given by \( \mathbf{u} \cdot \mathbf{v} = ac + bd \).

Step 2: Identify Components of Vectors

Identify the components of vectors \( \mathbf{u} \) and \( \mathbf{v} \).\( \mathbf{u} = 4\mathbf{i} + 0\mathbf{j} \) gives components \( a = 4 \), \( b = 0 \).\( \mathbf{v} = -1\mathbf{i} + 3\mathbf{j} \) gives components \( c = -1 \), \( d = 3 \).

Step 3: Calculate the Dot Product

Compute the dot product using the formula: \( \mathbf{u} \cdot \mathbf{v} = ac + bd = 4(-1) + 0(3) \).

Step 4: Solve the Dot Product

Calculate \( 4(-1) + 0(3) = -4 + 0 = -4 \).

Step 5: Determine Perpendicularity

Since the dot product \( -4 \) is not equal to zero, the vectors \( \mathbf{u} \) and \( \mathbf{v} \) are not perpendicular.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics. It combines two vectors to produce a single number, often called a scalar. Calculating it involves multiplying corresponding components of two vectors and summing these products. If you have vectors \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{v} = c\mathbf{i} + d\mathbf{j} \), then their dot product is given by:
\[\mathbf{u} \cdot \mathbf{v} = ac + bd\]
The dot product is crucial in determining the relationship between two vectors. An important property of the dot product is determining perpendicularity. Remember, if two vectors are perpendicular, their dot product equals zero.
This property is because perpendicular vectors have orthogonal directions; hence their contribution to each other's direction cancels out. Calculating zero emphasizes these directional differences.

Vector Components

Understanding vector components is key to performing operations like the dot product. A vector in a plane consists of horizontal and vertical components often represented alongside unit vectors \( \mathbf{i} \) and \( \mathbf{j} \).
For example, the vector \( \mathbf{u} = 4\mathbf{i} + 0\mathbf{j} \) can be split into:

• 4 units in the direction of \( \mathbf{i} \) which is the horizontal component.
• 0 units in the direction of \( \mathbf{j} \), so there's no vertical component.

These components highlight not only the vector's direction but also its magnitude in specific dimensions.
For \( \mathbf{v} = -1\mathbf{i} + 3\mathbf{j} \), the components are \(-1\) in the \( \mathbf{i} \) direction and \(3\) in the \( \mathbf{j} \) direction.
The orientation of vector arrows on a graph can reinforce understanding of these components and demonstrate differences when comparing distinct vectors.

Perpendicularity of Vectors

Vectors being perpendicular is synonymous with orthogonality. This concept indicates they meet at a right angle (90°), and algebraically, this is verified via the dot product.
If the dot product of two vectors yields zero, the vectors do not influence each other's directions and are said to be perpendicular.

• This means perpendicular vectors share no component overlap, forming an L-shape.
• This property can be useful in physics and engineering applications to delineate force directions or indicate precise orientation.

In the given example, the calculated dot product of vectors \( \mathbf{u} \) and \( \mathbf{v} \) resulted in \(-4\). Therefore, these vectors are not perpendicular as the product isn't zero.
Recognizing the perpendicularity condition enhanced by computations like the dot product helps unravel vector interactions in various domains, leading to deeper insights into their spatio-mathematical behaviors.