Question

Find each scalar multiple of \(v\) and sketch its graph. \(\mathbf{v}=\langle 2,-2,1\rangle\) (a) - \(\mathbf{v}\) (b) \(2 \mathbf{v}\) (c) \(\frac{1}{2} \mathbf{v}\) (d) \(\frac{5}{2} \mathbf{v}\)

Short answer

The scalar multiples are: \n(a) -v = \(\langle -2, 2, -1 \rangle\), \n(b) 2v = \(\langle 4, -4, 2 \rangle\), \n(c) \(\frac{1}{2}v\) = \(\langle 1, -1, 0.5 \rangle\), \n(d) \(\frac{5}{2}v\) = \(\langle 5, -5, 2.5 \rangle\)

Step-by-step solution

Find -v

To find -v, multiply each component of \(v\) by -1: \n-1*2=-2, -1*(-2) = 2, -1*1=-1\nThus, -v = \(\langle -2, 2, -1 \rangle\)

Find 2v

To find 2v, multiply each component of \(v\) by 2: \n2*2=4, 2*(-2)=-4, 2*1=2\nThus, 2v = \(\langle 4, -4, 2 \rangle\)

Find \(\frac{1}{2}v\)

To find \(\frac{1}{2}v\), multiply each component of \(v\) by \(\frac{1}{2}\): \n\(\frac{1}{2}*2=1\), \(\frac{1}{2}*(-2)=-1\), \(\frac{1}{2}*1=0.5\)\nThus, \(\frac{1}{2}v\) = \(\langle 1, -1, 0.5 \rangle\)

Find \(\frac{5}{2}v\)

To find \(\frac{5}{2}v\), multiply each component of \(v\) by \(\frac{5}{2}\): \n\(\frac{5}{2}*2=5\), \(\frac{5}{2}*(-2)=-5\), \(\frac{5}{2}*1=2.5\)\nThus, \(\frac{5}{2}v\) = \(\langle 5, -5, 2.5 \rangle\)

Sketching the graphs

Using graphing tools available in mathematics software or even a 3D sketch on paper, sketch each of these vectors starting from the origin (0,0,0) to the point defined by the vector. These vector representations are in the form (x,y,z), so -v would start at the origin and end at the point (-2, 2, -1), 2v will end at (4, -4, 2), and so on.

Key concepts

Linear Algebra

Linear algebra is a fundamental field of mathematics that deals with the study of vectors, vector spaces, linear mappings, and systems of linear equations. It has applications in various areas of science and engineering, helping us understand multidimensional structures and solve complex problems mathematically. A vital operation in linear algebra is the concept of scalar multiplication.
In scalar multiplication, a vector is multiplied by a scalar (a single, real number), altering its magnitude, but not its direction unless the scalar is negative in which case it also reverses its direction. For example, given a vector \( \mathbf{v} = \langle 2, -2, 1 \rangle \) and a scalar \( k \), scalar multiplication produces a new vector \( k\mathbf{v} \). This is done by multiplying the scalar with each component of the vector, as shown in the original exercise problem.

Vector Operations

Vector operations include basic actions such as addition, subtraction, scalar multiplication, dot products, and cross products. These operations allow for the manipulation of vectors for practical objectives, such as computations in physics, computer graphics, and optimization problems. Scalar multiplication in particular changes the scale of a vector. For instance, if the initial vector represents a certain amount of force applied in a direction, scalar multiplication by 2 would double the force, while a scalar of \( \frac{1}{2} \) would halve it, without changing the original direction of application.

Graphing Vectors

Graphing vectors is an essential visual tool to understand vector operations better. When a vector in 3D space is represented, such as \( \mathbf{v} \), it’s graphically depicted as an arrow pointing from the origin to a point in space. The vector's direction is the path of this arrow, and its magnitude is shown by the length of the arrow. When scales such as -1, 2, \( \frac{1}{2} \) and \( \frac{5}{2} \) are applied, the resultant vectors can be visualized as differently scaled arrows in space, either shorter or longer based on the magnitude of the scalar. The precise end-point of these arrows can be determined by performing scalar multiplication on the vector's components.

3D Coordinate System

The 3D coordinate system extends the familiar 2D Cartesian plane into three dimensions, introducing a z-axis perpendicular to both the x- and y-axes. This system allows us to represent the position of points in space, and by extension, vectors. The vector \( \langle x, y, z \rangle \) denotes a point in space, reachable from the origin (0, 0, 0) by following the vector's direction. The scalar multiples of a vector in the 3D coordinate system help us visualize how magnitudes affect vectors within this volumetric space, such as stretching or compressing along certain dimensions, as seen with the scalar multiples of vector \( \mathbf{v} \).