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Chapter 8: Problem 531

For the following exercises, given \(v, \quad \operatorname{draw} v, 3 v\) and \(\frac{1}{2} v\). $$\langle- 1,4\rangle$$

Short Answer

Expert verified
\(v = \langle -1, 4 \rangle\); draw a vector ending at (-1, 4); \(3v = \langle -3, 12 \rangle\); \(\frac{1}{2}v = \langle -\frac{1}{2}, 2 \rangle\).

Step by step solution

01

Understanding Vector Operations

We are given the vector \( v = \langle -1, 4 \rangle \). The tasks are to find the vector \( v \), draw \( v \), compute \( 3v \), and \( \frac{1}{2}v \).
02

Vector Representation

The given vector \( v = \langle -1, 4 \rangle \) is a two-dimensional vector where the first component is \( -1 \) and the second component is \( 4 \). This means that \( v \) points one unit to the left and four units up in the 2D coordinate system.
03

Drawing the Vector

To draw the vector \( v = \langle -1, 4 \rangle \), start at the origin \((0, 0)\) of a 2D coordinate plane. Then move left to \( -1 \) along the x-axis and up to \( 4 \) along the y-axis. The point at \( (-1, 4) \) is the tip of the vector, and you can draw an arrow from the origin to this point to represent \( v \).
04

Calculating Scalar Multiplication 3v

To find \( 3v \), multiply each component of \( v = \langle -1, 4 \rangle \) by 3. This gives:\[ 3v = 3 \times \langle -1, 4 \rangle = \langle 3(-1), 3(4) \rangle = \langle -3, 12 \rangle \]
05

Calculating Scalar Multiplication 1/2v

To find \( \frac{1}{2} v \), multiply each component of \( v = \langle -1, 4 \rangle \) by \( \frac{1}{2} \). This gives:\[ \frac{1}{2} v = \frac{1}{2} \times \langle -1, 4 \rangle = \langle \frac{1}{2}(-1), \frac{1}{2}(4) \rangle = \langle -\frac{1}{2}, 2 \rangle \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scalar Multiplication
Scalar multiplication involves multiplying a vector by a scalar (a single number). This operation stretches or shrinks the vector by the scalar's factor. It means each component of the vector is multiplied by the scalar.
  • Formula: For a vector \( v = \langle x, y \rangle \) and a scalar \( c \), the scalar multiplication is \( c \times v = \langle c \times x, c \times y \rangle \).
  • Example: With \( v = \langle -1, 4 \rangle \) and scalars 3 and \( \frac{1}{2} \), we get \( 3v = \langle -3, 12 \rangle \) (vector is stretched) and \( \frac{1}{2} v = \langle -\frac{1}{2}, 2 \rangle \) (vector is shrunk).
Though simple, mastering scalar multiplication is vital in many fields, such as physics for scaling forces or velocities.
Two-Dimensional Vector
A two-dimensional vector has two components and can be represented as an ordered pair, typically in the format \( \langle x, y \rangle \). Each component corresponds to the position on a coordinate plane.
  • X-component: Represents the horizontal displacement.
  • Y-component: Represents the vertical displacement.
For example, the vector \( v = \langle -1, 4 \rangle \), has an x-component of -1, indicating it moves 1 unit left, and a y-component of 4, indicating it moves 4 units up. Understanding these basic directions is crucial as it forms the basis of more complex operations.
Coordinate Plane
The coordinate plane is a two-dimensional surface where each point is uniquely determined by an ordered pair of numbers. It's divided into four quadrants by two perpendicular lines: the x-axis (horizontal) and y-axis (vertical). This plane is essential for graphing and visualizing vectors.
  • Origin: The center of the plane, denoted as (0,0).
  • Quadrants: Four sections separated by the axes, labeled I to IV.
In the context of our problem, to draw the vector \( v = \langle -1, 4 \rangle \), we begin at the origin. The instruction, left by 1 and up by 4, leads us to the vector's endpoint in the coordinate plane. This graphical representation helps in visualizing magnitude and direction.
Vector Representation
Vectors can be represented using different notations. Often, vectors are shown graphically as arrows from one point to another in a coordinate plane.
  • Geometric Representation: An arrow with direction and magnitude (length).
  • Component Form: Written as \( \langle x, y \rangle \), showing horizontal and vertical impacts separately.
For example, the vector \( \langle -1, 4 \rangle \) entails moving from the origin to point (-1,4), with the direction from (0,0) to the tip showing its effect. Vectors allow us to compactly express and handle quantities that have both magnitude and direction, fundamental in modeling real-world phenomena, like motion.

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