Problem 28 - Precalculus

Question

Determine whether each pair of vectors is orthogonal. $$\langle 8,3\rangle \text { and }\langle-6,16\rangle$$

Step-by-Step Solution

Answer

The vectors are orthogonal.

Step 1: Understand Orthogonality

Two vectors are orthogonal if their dot product is zero. We need to calculate the dot product of \( \langle 8, 3 \rangle \) and \( \langle -6, 16 \rangle \) to determine if they are orthogonal.

Step 2: Calculate the Dot Product

The dot product of two vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is computed as \( a \cdot c + b \cdot d \). For the vectors \( \langle 8, 3 \rangle \) and \( \langle -6, 16 \rangle \), this calculation is: \[ 8 \times (-6) + 3 \times 16 \].

Step 3: Perform the Multiplication

Calculate each part of the dot product: \( 8 \times (-6) = -48 \) and \( 3 \times 16 = 48 \).

Step 4: Sum the Results

Add the results from the multiplications: \( -48 + 48 = 0 \).

Step 5: Conclude on Orthogonality

Since the dot product is 0, the vectors \( \langle 8, 3 \rangle \) and \( \langle -6, 16 \rangle \) are orthogonal.

Key Concepts

Dot Product and Orthogonality

The dot product is an essential concept in linear algebra. It helps determine if two vectors are orthogonal. Orthogonality is a fancy way of saying the vectors are perpendicular to each other. Two vectors are orthogonal if their dot product equals zero. The dot product of two vectors \( \langle a, b \rangle \) and \( \langle c, d \rangle \) is found by multiplying their corresponding components and then adding the results. This is done through the formula:

• First, multiply the first components together: \( a \cdot c \).
• Next, multiply the second components together: \( b \cdot d \).
• Finally, add these products together: \( a \cdot c + b \cdot d \).

In our exercise, when calculating the dot product for the vectors \( \langle 8, 3 \rangle \) and \( \langle -6, 16 \rangle \), we get \(-48 + 48 = 0\). The dot product is zero, confirming that these vectors are indeed orthogonal.

Vector Multiplication

Vector multiplication comes in different forms, with the dot product being one of them. It’s important to recognize the role this multiplication plays in making calculations easy and meaningful.
The basic steps for vector multiplication using the dot product include:

• Identify the components of each vector. In our example, these are \(8, 3\) for the first vector and \(-6, 16\) for the second one.
• Multiply each corresponding component. This means multiplying 8 by -6 and 3 by 16.
• Add these products to get the dot product.

A significant outcome from vector multiplication, using the dot product, is the determination of orthogonality, as shown in our example. Whether or not vectors are orthogonal is a key focus in many mathematical problems and practical applications.

Properties of Vectors and their Impact

Vectors have unique properties that can alter the calculations and outcomes in various mathematical scenarios. Here are several vital properties:

• Additive Properties: Vectors can be added together, and this addition is commutative, meaning \( \langle a, b \rangle + \langle c, d \rangle = \langle c, d \rangle + \langle a, b \rangle \).
• Scalar Multiplication: You can multiply a vector by a scalar, which is a simple number. This changes the vector’s magnitude but not its direction (except if the scalar is negative, which reverses direction).
• Dot Product Zero Property: If the dot product of two vectors is zero, the vectors are orthogonal.

These properties are crucial for understanding how vectors interact and transform. By recognizing these properties, you can solve tasks efficiently, not just in theoretical exercises, but also in real-world situations where vectors are used.