Question

In Exercises 49 and \(50,\) find each scalar multiple of \(v\) and sketch its graph. \(\mathbf{v}=\langle 1,2,2\rangle\) (a) \(2 \mathbf{v}\) (b) \(-\mathbf{v}\) (c) \(\frac{3}{2} \mathbf{v}\) (d) \(0 \mathbf{v}\)

Short answer

The scalar multiples of vector \(v = \langle 1,2,2 \rangle\) are: (a) \(2v = \langle 2,4,4 \rangle\), (b) \(-v = \langle -1,-2,-2 \rangle\), (c) \(\frac{3}{2}v = \langle 1.5,3,3 \rangle\), and (d) \(0v = \langle 0,0,0 \rangle\). Each of these vectors represented graphically will scale or reflect the original vector \(v\) in different ways.

Step-by-step solution

Find Scalar Multiples

First, calculate the scalar multiples of vector \(v\). (a) \(2v = 2 \cdot \langle 1,2,2 \rangle = \langle 2,4,4 \rangle\)(b) \(-v = -1 \cdot \langle 1,2,2 \rangle = \langle -1,-2,-2 \rangle\)(c) \(\frac{3}{2}v = \frac{3}{2} \cdot \langle 1,2,2 \rangle = \langle 1.5,3,3 \rangle\)(d) \(0v = 0 \cdot \langle 1,2,2 \rangle = \langle 0,0,0 \rangle\)

Sketch the Graphs

Next, represent these vectors graphically. The graph should be in 3D as the vectors are three-dimensional. (a) Vector \(2v = \langle 2,4,4 \rangle\) is twice as long as vector \(v\), but in the same direction.(b) Vector \(-v = \langle -1,-2,-2 \rangle\) is equal in length to vector \(v\), but in the opposite direction.(c) Vector \(\frac{3}{2}v = \langle 1.5,3,3 \rangle\) is 1.5 times as long as \(v\), but in the same direction.(d) Vector \(0v = \langle 0,0,0 \rangle\) is the zero vector, which is just a point at the origin.

Key concepts

Vector Operations

Vector operations are fundamental in various branches of mathematics, specifically linear algebra. To perform scalar multiplication, like in the given exercise, you multiply each component of the vector by the scalar value.

For instance, when you see an operation such as \(2\mathbf{v}\), it implies you should multiply each component of the vector \(\mathbf{v}\) by 2. In the case of vector \(\mathbf{v}=\langle 1,2,2 \rangle\), the scalar multiplication by 2 yields \(\langle 2,4,4 \rangle\), meaning each dimension of the original vector is doubled.

Contrastingly, negative scalar multiplication, like \( -\mathbf{v} \), inverts the direction of the vector. Using a fractional scalar, such as \(\frac{3}{2}\mathbf{v}\), scales the vector proportionally without changing its direction. And finally, multiplying any vector by zero results in a zero vector \(\mathbf{0}\), which has no direction or magnitude and is represented by the origin point in a graph.

Graphing Vectors in 3D

Graphing vectors in three dimensions provides a visual understanding of their direction and magnitude. A 3D vector has three components that correspond to its projections along the x, y, and z axes, which are graphically represented in a three-dimensional space.

For the vectors from our exercise, each one can be plotted starting at the origin (0,0,0) and extending to the point represented by its coordinates. For example, the scalar multiple \(2\mathbf{v}\) would extend twice as far in each dimension compared to the original vector \(\mathbf{v}\).

In 3D graphing, accurately depicting the length and direction of a vector is crucial, as it conveys valuable information about the vector's properties. For instance, \(\mathbf{-v}\) would still start at the origin but extend in the exact opposite direction from \(\mathbf{v}\).

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and vector operations. Scalar multiplication, one of these operations, is an integral concept of linear algebra.

Through linear algebra, we can solve system of equations, perform transformations, and even analyze rotations in space which are crucial for advanced fields such as engineering, physics, and economics. Understanding how to manipulate vectors through operations like scalar multiplication is foundational in linear algebra.

The exercise provided offers a concrete example of how vectors are scaled within a vector space. When the vector is multiplied by different scalar values, we observe distinct outcomes that align with linear algebra theories.

Vector Algebra

Vector algebra is the field that deals with calculations involving vectors. Unlike traditional algebra which deals with scalars (just numbers), vector algebra includes both magnitude and direction in its operations.

The operations involved in vector algebra can include addition, subtraction, and as seen in our exercise, scalar multiplication. Each of these operations follows specific rules that preserve the vector properties such as distributive and associative properties.

In the context of the exercise, one can appreciate how these operations alter a vector. By understanding the principles of vector algebra, we see the real-world effects of these operations, such as changing the force applied to an object (magnitude) or the direction of an object's motion.