Problem 30 - Precalculus

Question

Determine whether each pair of vectors is orthogonal. $$\langle-7,3\rangle \text { and }\left\langle\frac{1}{7},-\frac{1}{3}\right\rangle$$

Step-by-Step Solution

Answer

The vectors are not orthogonal.

Step 1: Understand what it means for vectors to be orthogonal

Two vectors are orthogonal if their dot product is zero. For vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \), the dot product is calculated as \( a \cdot c + b \cdot d \).

Step 2: Identify vector components

The first vector is \( \langle -7, 3 \rangle \) and the second vector is \( \left\langle \frac{1}{7}, -\frac{1}{3} \right\rangle \). The components for each vector are: - For the first vector: \( a = -7 \), \( b = 3 \).- For the second vector: \( c = \frac{1}{7} \), \( d = -\frac{1}{3} \).

Step 3: Calculate the dot product

Substitute the components into the dot product formula:\[a \cdot c + b \cdot d = (-7) \cdot \frac{1}{7} + 3 \cdot \left(-\frac{1}{3}\right)\]Simplify the expression:\[= -1 + (-1) = -2\]

Step 4: Determine orthogonality

The dot product calculated is \(-2\), which is not equal to zero. Therefore, the vectors \( \langle -7, 3 \rangle \) and \( \left\langle \frac{1}{7}, -\frac{1}{3} \right\rangle \) are not orthogonal.

Key Concepts

Dot Product

The dot product is a fundamental tool used in vector mathematics to determine relationships between vectors. It enables us to find how aligned two vectors are in a given space. To compute the dot product of two vectors, imagine you have vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \). The dot product formula is written as:\[ a \cdot c + b \cdot d \]This calculation multiplies corresponding components of the vectors and then sums these products.

• In the given exercise, for vectors \( \langle -7, 3 \rangle \) and \( \left\langle \frac{1}{7}, -\frac{1}{3} \right\rangle \), the dot product was calculated as: \(-7 \cdot \frac{1}{7} + 3 \cdot \left(-\frac{1}{3}\right)\).
• This expression then simplifies to \(-1 + (-1) = -2\).

The result of \(-2\) indicates that these vectors aren't orthogonal because the dot product isn't zero. Understanding the dot product helps solve diverse vector-based problems, making it essential to grasp this concept fully.

Vector Components

Every vector in a two-dimensional space can be broken down into its components. These components essentially represent how much the vector extends in each axis direction. Regarding a vector \( \langle a, b \rangle \):

• \(a\) refers to the component along the x-axis.
• \(b\) refers to the component along the y-axis.

Similarly, the vector \( \langle c, d \rangle \) represents its components along the x and y axes as \(c\) and \(d\) respectively. In this exercise:- The first vector is \( \langle -7, 3 \rangle \). So, \(a = -7\) and \(b = 3\).- The second vector is \( \left\langle \frac{1}{7}, -\frac{1}{3} \right\rangle \). Here, \(c = \frac{1}{7}\) and \(d = -\frac{1}{3}\).The knowledge of vector components is vital for carrying out calculations such as the dot product as well as for understanding how vectors influence movement and direction in space.

Orthogonality Condition

The orthogonality condition is a crucial concept when examining the relationship between vectors. Vectors are defined as orthogonal if their dot product equals zero.

• Orthogonal vectors signify that the vectors are perpendicular to each other.
• If the vectors align perfectly at a right angle, they do not share any component in any direction.

Using the given example, once the dot product was calculated to be \(-2\), this indicated that the vectors \( \langle -7, 3 \rangle \) and \( \left\langle \frac{1}{7}, -\frac{1}{3} \right\rangle \) are not orthogonal since their product isn't zero. Understanding the orthogonality condition enables insight into how vectors interact, important for solving problems in physics and engineering.