Problem 16 - Algebra and Trigonometry

Question

Determine whether the given vectors are perpendicular. $$ \mathbf{u}=\langle 0,-5\rangle, \quad \mathbf{v}=\langle 4,0\rangle $$

Step-by-Step Solution

Answer

The vectors are perpendicular.

Step 1: Recall the Perpendicular Condition

Two vectors are perpendicular if and only if their dot product is zero. This means for vectors \( \mathbf{u} \) and \( \mathbf{v} \), \( \mathbf{u} \cdot \mathbf{v} = 0 \).

Step 2: Calculate the Dot Product

The dot product of two vectors \( \mathbf{u} = \langle a, b \rangle \) and \( \mathbf{v} = \langle c, d \rangle \) is calculated as: \( \mathbf{u} \cdot \mathbf{v} = a \cdot c + b \cdot d \). For the given vectors, \( \mathbf{u} = \langle 0, -5 \rangle \) and \( \mathbf{v} = \langle 4, 0 \rangle \), this becomes: \[ \mathbf{u} \cdot \mathbf{v} = 0 \cdot 4 + (-5) \cdot 0. \]

Step 3: Simplify the Dot Product Expression

Simplify the expression: \[ 0 \cdot 4 + (-5) \cdot 0 = 0 + 0 = 0. \] This confirms that the dot product is zero.

Step 4: Conclude Perpendicularity

Since the dot product of \( \mathbf{u} \) and \( \mathbf{v} \) is zero, the vectors are perpendicular. Thus, \( \mathbf{u} \) and \( \mathbf{v} \) are perpendicular vectors.

Key Concepts

The Dot Product

The dot product is a fundamental operation in vector mathematics that combines two vectors to produce a single scalar value. It is denoted by a dot between two vectors, for example, \( \mathbf{u} \cdot \mathbf{v} \). Calculating the dot product helps us understand the directional relationship between vectors. The basic formula for the dot product of vectors \( \mathbf{u} = \langle a, b \rangle \) and \( \mathbf{v} = \langle c, d \rangle \) is given by:

• \( \mathbf{u} \cdot \mathbf{v} = a \cdot c + b \cdot d \)

In our exercise, this formula helped us determine that the vectors \( \mathbf{u} = \langle 0, -5 \rangle \) and \( \mathbf{v} = \langle 4, 0 \rangle \) have a dot product of 0, because:

• \( 0 \cdot 4 + (-5) \cdot 0 = 0 \)

When the dot product results in zero, it reveals something special about the vectors: they are perpendicular.

Understanding Vector Mathematics

Vector mathematics is an essential part of mathematics that involves the use of vectors, which are quantities that have both magnitude and direction. Vectors can be represented geometrically as arrows or numerically with sets of coordinates like \( \langle a, b \rangle \). Here are some key concepts of vector mathematics:

• Vectors have both direction and magnitude.
• They are often visualized as arrows pointing in space.
• Numeric representation in two dimensions uses pairs like \( \langle a, b \rangle \).
• Operations like addition, scalar multiplication, and dot product are common.

In our exercise, we dealt with vector operations using the dot product. This process involved breaking down the components of the vectors into their respective coordinates \( a, b \) and \( c, d \) to calculate how they function together. Understanding these basics allows us to assess vector behaviors like directionality and perpendicularity.

Vector Perpendicularity

Vector perpendicularity is a noteworthy concept in vector mathematics, describing when two vectors meet at right angles (90 degrees). This perpendicular position is crucial as it signifies no component of one vector points along the direction of the other.

• Vectors are perpendicular when their dot product equals zero.
• Such a relationship implies they form an orthogonal pair.
• An orthogonal pair aligns at a 90-degree angle.

For the vectors \( \mathbf{u} \) and \( \mathbf{v} \) in our problem, we used the dot product to confirm their perpendicularity. With the simple calculation showing \( 0 \cdot 4 + (-5) \cdot 0 = 0 \), we determined that these two vectors are indeed perpendicular. This means \( \mathbf{u} \) lies entirely along one axis, while \( \mathbf{v} \) follows another, uniquely independent direction.