Problem 16 - 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. $$-2 c+2 d$$

Step-by-Step Solution

Answer

The result is \(\langle -16, 2 \rangle\).

Step 1: Scale Vector c

Given vector \( \mathbf{c} = \langle 6, -1 \rangle \), scale it by \(-2\). The result is: \(-2 \cdot \mathbf{c} = -2 \cdot \langle 6, -1 \rangle = \langle -12, 2 \rangle\).

Step 2: Scale Vector d

Given vector \( \mathbf{d} = \langle -2, 0 \rangle \), scale it by \(2\). The result is: \(2 \cdot \mathbf{d} = 2 \cdot \langle -2, 0 \rangle = \langle -4, 0 \rangle\).

Step 3: Add Scaled Vectors

Add the results of the scaled vectors from Steps 1 and 2: \(\langle -12, 2 \rangle + \langle -4, 0 \rangle = \langle -12 + (-4), 2 + 0 \rangle = \langle -16, 2 \rangle\).

Key Concepts

Scaling Vectors

Scaling vectors is like stretching or shrinking them. Imagine a vector as an arrow in a two-dimensional plane that shows both a direction and a length (magnitude). To scale a vector, you multiply each of its components by a number, known as the scalar. This action changes the length of the vector without altering its direction, unless you scale by a negative number, which also inverts the direction.

For example, if we have a vector \( \mathbf{v} = \langle x, y \rangle \) and we scale it by a scalar \( k \), the new vector is \( k \cdot \mathbf{v} = \langle kx, ky \rangle \).

• Scaling by a positive number keeps the direction the same but changes the length.
• Scaling by a negative number changes both the direction and the length.

In our exercise, vector \( \mathbf{c} = \langle 6, -1 \rangle \) was scaled by \(-2\), resulting in \( \langle -12, 2 \rangle \). Similarly, vector \( \mathbf{d} = \langle -2, 0 \rangle \) was scaled by \(2\), resulting in \( \langle -4, 0 \rangle \). This shows how each component of the vectors is multiplied by the scalar to achieve the scaling.

Vector Addition

Vector addition is the process of combining two or more vectors to get a new vector. The method is straightforward: you add corresponding components together.

For instance, if we have two vectors \( \mathbf{v} = \langle v_1, v_2 \rangle \) and \( \mathbf{u} = \langle u_1, u_2 \rangle \), their sum \( \mathbf{v} + \mathbf{u} \) is given by \( \langle v_1 + u_1, v_2 + u_2 \rangle \).

• Add the x-components from each vector.
• Add the y-components from each vector.

This operation yields a new vector that represents the cumulative effect of the original vectors – almost like combining forces. In the exercise, after the vectors were scaled, they were added together: \( \langle -12, 2 \rangle + \langle -4, 0 \rangle = \langle -16, 2 \rangle \). The simplicity of adding corresponding coordinates helps visualize the movement or force in coordinate geometry.

Linear Combinations

A linear combination involves using scaling and addition together to form a new vector from existing ones. This is a vital concept in linear algebra and vector geometry.

Suppose you have vectors \( \mathbf{u} \) and \( \mathbf{v} \), and scalars \( a \) and \( b \). The linear combination is expressed as \( a\mathbf{u} + b\mathbf{v} \). This combines the effects of scaling vectors with scalar multiplication and then adding them together to create a new vector.

Generally, a linear combination can be used to express any vector in a plane if enough independent vectors are given. This is powerful in terms of determining spans and understanding dimensions. In our specific exercise, we have formed a linear combination of vectors \( \mathbf{c} \) and \( \mathbf{d} \) by scaling them with \(-2\) and \(2\), respectively, and then adding the results: \(-2\mathbf{c} + 2\mathbf{d} = \langle -16, 2 \rangle\). Linear combinations are essential to form new vectors and determine relationships within vector spaces.

Coordinate Geometry

Coordinate geometry, sometimes referred to as analytic geometry, is how we use algebra in geometry. It enables us to describe geometrical shapes like lines, segments, and shapes using a coordinate system.

Vectors are integral to coordinate geometry as they describe positions and directions within this system. In two-dimensional coordinate systems, such as the Cartesian plane, each point is defined by a pair of coordinates \( (x, y) \). Vectors help in translating and transforming these points across the plane.

• Use vector operations to define and solve geometric problems.
• Visualize vectors as arrows pointing from the origin \((0, 0)\) to a point \((x, y)\).

In the exercise, vectors \( \mathbf{c} \) and \( \mathbf{d} \) are represented as points on this plane with their corresponding coordinates. The operations of scaling and addition then navigate this space, altering the vectors’ lengths and directions, or combining them to yield new locations or movements within the plane. Coordinate geometry thus provides the framework for understanding the geometric implications and transformations involving vectors.