Problem 21 - Precalculus

Question

Assume that the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c},\) and \(\mathbf{d}\) are defined as follows: $$\mathbf{a}=\langle 2,3\rangle \quad \mathbf{b}=\langle 5,4\rangle \quad \mathbf{c}=\langle 6,-1\rangle \quad \mathbf{d}=\langle-2,0\rangle$$ Compute each of the indicated quantities. $$\mathbf{a}+(\mathbf{b}+\mathbf{c})$$

Step-by-Step Solution

Answer

\( \langle 13, 6 \rangle \)

Step 1: Understand Vector Addition

Vector addition is performed by adding corresponding components. If you have two vectors \( \mathbf{x} = \langle x_1, x_2 \rangle \) and \( \mathbf{y} = \langle y_1, y_2 \rangle \), then their sum is \( \mathbf{x} + \mathbf{y} = \langle x_1 + y_1, x_2 + y_2 \rangle \).

Step 2: Add Vectors \( \mathbf{b} \) and \( \mathbf{c} \)

Given \( \mathbf{b} = \langle 5, 4 \rangle \) and \( \mathbf{c} = \langle 6, -1 \rangle \), we add them to get: \( \mathbf{b} + \mathbf{c} = \langle 5 + 6, 4 - 1 \rangle = \langle 11, 3 \rangle \).

Step 3: Add Result to Vector \( \mathbf{a} \)

Now, add the resulting vector \( \langle 11, 3 \rangle \) to \( \mathbf{a} = \langle 2, 3 \rangle \): \( \mathbf{a} + (\mathbf{b} + \mathbf{c}) = \langle 2 + 11, 3 + 3 \rangle = \langle 13, 6 \rangle \).

Key Concepts

Vectors

Vectors are mathematical objects that have both magnitude and direction. They are often visually represented as arrows, where the length of the arrow signifies the vector's magnitude and the arrow's orientation indicates the direction. Vectors are used extensively in physics and engineering to represent quantities such as force, velocity, and displacement.

Notably, a vector differs from a scalar, which only has magnitude without direction. For example, 5 meters is a scalar, whereas a displacement of 5 meters east is a vector. Vectors can be represented in two or three-dimensional space, and each dimension is associated with its corresponding components.

• In physics, vectors are crucial for understanding movement and forces.
• They allow us to describe how objects move and interact in space.
• Vectors help in visualizing and computing phenomena occurring in more than one dimension.
• Key properties of vectors include magnitude (length) and direction (orientation).

Vectors in Two Dimensions

When we discuss vectors in two dimensions, we are referring to vectors that lie in a plane and are expressed using two components, usually denoted as \( \langle x, y \rangle \). Here, "\(x\)" and "\(y\)" represent the components of the vector along the horizontal and vertical directions, respectively.

Two-dimensional vectors are particularly useful in fields like physics and engineering where majority of the movements or forces usually occur on a flat surface.

• Commonly represented using ordered pairs like \( \langle a, b \rangle \).
• They allow for the description of planar movements, such as those parallel to the earth's surface.
• Two-dimensional vectors are fundamental in analyzing objects that may be moving in directions, such as on a road map or within a coordinate system.
• Combining two-dimensional vectors requires adding their respective components.

Vector Operations

Vector operations include a variety of actions such as addition, subtraction, and scalar multiplication, all of which are important when working with vectors in mathematics and physics.

**Vector Addition**
To add vectors together, simply add their corresponding components. For example, given vectors \(\mathbf{x} = \langle x_1, x_2 \rangle\) and \(\mathbf{y} = \langle y_1, y_2 \rangle\), the sum is \(\mathbf{x} + \mathbf{y} = \langle x_1 + y_1, x_2 + y_2 \rangle\). This is called component-wise addition. It's a straightforward process, simplifying many physics and geometry problems.

Steps for Addition:

• Add the first components: \(x_1 + y_1\).
• Add the second components: \(x_2 + y_2\).
• Construct the resulting vector using these sums.

The result is a new vector that combines the effects of the original vectors. Other operations, like subtraction and scalar multiplication, follow similar principles, where the operation is applied individually to the vector's components.