Problem 62 - Precalculus with Calculus Previews

Question

In Problems \(49-62, \mathbf{u}=\langle 2,-3\rangle, \mathbf{v}=\langle-1,5\rangle,\) and \(\mathbf{w}=\langle 3,-2\rangle .\) Find the indicated scalar or vector. $$ (\mathbf{w} \cdot \mathbf{v}) \mathbf{u} $$

Step-by-Step Solution

Answer

The result is the vector \( \langle -26, 39 \rangle \).

Step 1: Understanding Dot Product

The dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is calculated as \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \). We need to find the dot product \( \mathbf{w} \cdot \mathbf{v} \).

Step 2: Calculate \( \mathbf{w} \cdot \mathbf{v} \)

Given \( \mathbf{w} = \langle 3, -2 \rangle \) and \( \mathbf{v} = \langle -1, 5 \rangle \), compute the dot product: \( \mathbf{w} \cdot \mathbf{v} = 3(-1) + (-2)(5) = -3 - 10 = -13 \).

Step 3: Multiply Result by Vector \( \mathbf{u} \)

Now that we have \( \mathbf{w} \cdot \mathbf{v} = -13 \), we multiply this scalar by \( \mathbf{u} = \langle 2, -3 \rangle \). The result is a vector: \( -13 \cdot \mathbf{u} = -13 \cdot \langle 2, -3 \rangle = \langle -13 \cdot 2, -13 \cdot (-3) \rangle \).

Step 4: Perform Scalar Multiplication

Compute the final vector: \( \langle -13 \cdot 2, -13 \cdot (-3) \rangle = \langle -26, 39 \rangle \). Thus, the result is the vector \( \langle -26, 39 \rangle \).

Key Concepts

dot product

When you hear the term "dot product," think of it as a way to combine two vectors to produce a single number, also known as a scalar. This operation is fundamental in vector mathematics because it helps us understand how vectors relate to each other. Given two vectors, \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), the dot product is calculated by multiplying corresponding components and then adding the results. This is expressed as:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \)

For instance, if we have vectors \( \mathbf{w} = \langle 3, -2 \rangle \) and \( \mathbf{v} = \langle -1, 5 \rangle \), the dot product \( \mathbf{w} \cdot \mathbf{v} \) computes to:

• \( 3(-1) + (-2)(5) = -3 - 10 = -13 \)

The dot product tells us how much one vector goes in the direction of another and is often used in physics and engineering to project force and motion.

scalar multiplication

Scalar multiplication involves taking a vector and multiplying it by a real number, known as a scalar. This operation stretches or shrinks the vector, but keeps its direction the same unless the scalar is negative, which reverses the direction. This concept helps in scaling vectors up or down based on the context of the problem.To multiply a vector \( \mathbf{u} = \langle u_1, u_2 \rangle \) by a scalar \( k \), you calculate:

• \( k \cdot \mathbf{u} = \langle k \cdot u_1, k \cdot u_2 \rangle \)

For the given exercise, after finding the dot product of \( \mathbf{w} \) and \( \mathbf{v} \) as \(-13\), the problem requires multiplying this scalar with vector \( \mathbf{u} = \langle 2, -3 \rangle \):

• \( -13 \cdot \mathbf{u} = \langle -13 \cdot 2, -13 \cdot (-3) \rangle = \langle -26, 39 \rangle \)

This illustrates how the multiplication scales the vector's components by \(-13\), resulting in a new vector with transformed magnitude and possibly reversed direction, depending on the sign of the scalar.

vector arithmetic

Vector arithmetic includes operations like addition, subtraction, and scaling of vectors, which are essential for performing complex mathematical and physical calculations. Understanding these operations lays the foundation for manipulating vectors to solve various problems.

• Vector Addition: To add two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), you simply add their corresponding components: \( \mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2 \rangle \).
• Vector Subtraction: To subtract one vector from another, subtract their corresponding components: \( \mathbf{a} - \mathbf{b} = \langle a_1 - b_1, a_2 - b_2 \rangle \).
• Scalar Multiplication: As detailed earlier, this scales the vector by a scalar.

Each operation follows specific mathematical rules, maintaining the logical and predictable behavior of vectors. These operations are crucial when dealing with vector spaces in physics, engineering, computer graphics, and machine learning. Understanding the fundamentals of vector arithmetic is crucial, as it sets the stage for more advanced topics like vector spaces and eigenvalues.