Problem 53 - 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. $$ \mathbf{w} \cdot \mathbf{w} $$

Step-by-Step Solution

Answer

The dot product \(\mathbf{w} \cdot \mathbf{w}\) is 13.

Step 1: Understand the 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 given by the formula: \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2\). For a vector \(\mathbf{w} = \langle 3, -2 \rangle\), the dot product with itself is \(\mathbf{w} \cdot \mathbf{w} = 3 \times 3 + (-2) \times (-2)\).

Step 2: Calculate the Dot Product for \(\mathbf{w} \cdot \mathbf{w}\)

Substitute the components of \(\mathbf{w}\) into the dot product formula: \( \mathbf{w} \cdot \mathbf{w} = 3 \times 3 + (-2) \times (-2) \). Calculate each term separately: \(3 \times 3 = 9\) and \((-2) \times (-2) = 4\).

Step 3: Sum the products to find the result

Add the two products from Step 2: \( 9 + 4 = 13 \). Therefore, the dot product \(\mathbf{w} \cdot \mathbf{w}\) is 13.

Key Concepts

Vectors

Vectors are mathematical objects that have both a magnitude and a direction. Imagine an arrow where:

• The length of the arrow represents the magnitude.
• The direction in which the arrow points represents its direction.

Whether we are dealing with two-dimensional or three-dimensional space, vectors are fundamental in physics, engineering, and computer graphics.
In our exercise, we have vectors like \( \mathbf{w} = \langle 3, -2 \rangle \), which are given in component form. This simply means that we describe the vector in terms of its horizontal and vertical movements.
Understanding vectors is essential because they allow us to model and solve real-world problems like navigation, forces, and speeds.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, a real number that scales the vector's magnitude without changing its direction. Here's what happens:

• The vector's size is multiplied by the scalar.
• Only its magnitude changes, not its direction (unless the scalar is negative, which also reverses the direction).

For example, multiplying \( \mathbf{w} = \langle 3, -2 \rangle \) by a scalar \( k \), you get a new vector: \( k \cdot \mathbf{w} = \langle k \cdot 3, k \cdot (-2) \rangle \).
This concept allows us to adjust vectors to represent larger or smaller quantities, making it incredibly useful in scaling movements or forces in physical simulations.

Vector Components

Vector components are the building blocks of a vector. They represent the effect of a vector in each dimension independently, often referred to as the horizontal and vertical components, especially in two-dimensional space.

• The first component is along the x-axis (horizontal).
• The second component is along the y-axis (vertical).

For example, our vector \( \mathbf{w} = \langle 3, -2 \rangle \) has two components: 3 in the x-direction and -2 in the y-direction. The components are crucial in calculations of vector operations such as the dot product, which relies on component-wise multiplication to determine a resulting scalar value.
Understanding vector components allows you to resolve any vector into manageable parts, making complex problems more approachable and less overwhelming.