Problem 18 - Precalculus Mathematics for Calculus

Question

Determine whether the given vectors are perpendicular. $$\mathbf{u}=2 \mathbf{i}, \quad \mathbf{v}=-7 \mathbf{j}$$

Step-by-Step Solution

Answer

The vectors are perpendicular because their dot product is zero.

Step 1: Understanding Perpendicular Vectors

Two vectors are perpendicular if their dot product is zero. The dot product of two vectors \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{v} = c\mathbf{i} + d\mathbf{j} \) is \( \mathbf{u} \cdot \mathbf{v} = ac + bd \).

Step 2: Identifying Components of the Vectors

For \( \mathbf{u} = 2 \mathbf{i} \), the components are \( a=2 \) and \( b=0 \). For \( \mathbf{v} = -7 \mathbf{j} \), the components are \( c=0 \) and \( d=-7 \).

Step 3: Calculate the Dot Product

Substitute the components into the dot product formula: \[ \mathbf{u} \cdot \mathbf{v} = (2)(0) + (0)(-7) = 0 + 0 = 0 \]

Step 4: Conclusion

Since the dot product \( \mathbf{u} \cdot \mathbf{v} = 0 \), vectors \( \mathbf{u} \) and \( \mathbf{v} \) are perpendicular.

Key Concepts

Dot Product

In mathematics, the dot product is an important operation when dealing with vectors. It is a way to multiply two vectors, resulting in a scalar value—this means the result is just a number, not a vector. To calculate the dot product for vectors in two-dimensional space, you use their components.

• For two vectors \(\mathbf{u} = a\mathbf{i} + b\mathbf{j}\) and \(\mathbf{v} = c\mathbf{i} + d\mathbf{j} \), the dot product is given by the formula: \( \mathbf{u} \cdot \mathbf{v} = ac + bd \).
• To find out if two vectors are perpendicular, compute their dot product. If the result is zero, the vectors are perpendicular to each other.

This critical property of perpendicular vectors—having a dot product of zero—allows us to easily assess whether vectors are at right angles, which is invaluable in geometry and physics.

Vector Components

Vectors are often broken down into components to simplify calculations.

• Each vector can be represented by its components along the axis in a coordinate system. For example, a vector \( \mathbf{u} = 2 \mathbf{i} \) has a component of 2 in the direction of the horizontal axis often denoted by \( \mathbf{i} \).
• Similarly, \( \mathbf{v} = -7 \mathbf{j} \) has no component in the horizontal direction and a component of -7 in the vertical direction, denoted by \( \mathbf{j} \).
• These components are like the building blocks of the vector, as they tell you precisely how far and in what direction the vector extends along each axis.

Understanding vector components is essential for carrying out operations like the dot product, where each component is multiplied accordingly.

Vector Notation

Vector notation is a universal way to write vectors, making it easier to perform calculations and convey information. In standard vector notation,

• Vectors are indicated by boldface letters, such as \( \mathbf{u} \) and \( \mathbf{v} \).
• Unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) denote the directions of the horizontal and vertical axes. A vector \( \mathbf{u} = 2 \mathbf{i} \) means it goes twice as far in the direction of the \( \mathbf{i} \) unit vector.
• This notation clearly distinguishes between the magnitude and direction components of a vector, allowing simpler calculations such as adding vectors or finding the dot product.

By mastering vector notation, you unlock a powerful language used extensively in physics, engineering, and mathematics.