Ron Larson, Bruce Edwards
ISBN #9781285774770
6th Edition
8,973 Questions
Homework Questions
This chapter section introduces the concept of vectors in the plane and extends these ideas to three-dimensional space. Key topics include writing vectors in component form, computing magnitudes, performing vector operations like addition and scalar multiplication, normalizing vectors to create unit vectors, and representing vectors as linear combinations of standard unit vectors. The extension into 3D involves a clear understanding of the Cartesian coordinate system, its axes, and the right-handed orientation, which are essential for applications in physics, engineering, and computer graphics.
1
Write the component form of a vector and convert between a directed line segment and its coordinate representation.
2
Perform vector operations (addition, subtraction, scalar multiplication) and interpret the results geometrically.
3
Compute the magnitude of a vector and normalize it to form a unit vector.
4
Express any vector as a linear combination of the standard unit vectors i and j.
5
Understand the basics of three-dimensional coordinate systems and extend vector concepts to space.
CONCEPT
DEFINITION
Scalar
A quantity described by a single real number, representing magnitude without direction (e.g., mass, temperature).
Vector
A quantity that has both magnitude and direction, represented geometrically as a directed line segment.
Component Form
Representation of a vector by listing its horizontal and vertical components, typically as (v1, v2).
Magnitude (Length)
The distance between the initial and terminal points of a vector, calculated using the distance formula.
Unit Vector
A vector with a magnitude of 1 that indicates direction; can be found by dividing a vector by its magnitude.
Standard Unit Vectors
The fixed vectors i = (1, 0) and j = (0, 1) which form the basis for the plane and are used to express any vector as a linear combination.
Three-Dimensional Coordinate System
An extension of the Cartesian plane where a third axis (z-axis) is added, enabling representation of points as ordered triples (x, y, z).
Right-Handed System
A convention for orienting the three-dimensional coordinate system so that when the index finger points along the positive x-axis and the middle finger along the positive y-axis, the thumb points in the direction of the positive z-axis.
Sketching a Vector. (a) find the component form of the vector $v$ and $(b)$ sketch the vector with its initial point at the origin. (GRAPH CANNOT COPY)
Find the vectors u and v whose initial and terminal points are given. Show that u and v are equivalent. $$\begin{aligned} &\mathbf{u}:(3,2),(5,6)\\ &\mathbf{v}:(1,4),(3,8) \end{aligned}$$
QUESTION
Given an initial point P(3, 7) and terminal point Q(2, 5), write the vector in component form.
STEP-BY-STEP ANSWER:
Step 1: Identify the coordinates of P and Q. P = (3, 7) and Q = (2, 5). Step 2: Subtract the coordinates of P from Q: (2 - 3, 5 - 7) = (-1, -2). Step 3: Final Answer: The vector in component form is <-1, -2>. Final Answer:
Component Form of a Vector
For the vector v = <5, 12>, find its magnitude.
Step 1: Use the formula for magnitude: |v| = √(v1^2 + v2^2). Step 2: Compute: √(5^2 + 12^2) = √(25 + 144) = √169 = 13. Step 3: Final Answer: The magnitude of v is 13. Final Answer:
Magnitude of a Vector
Find the unit vector in the direction of v = <5, 12>.
Step 1: Determine the magnitude of v, which is 13. Step 2: Divide each component by 13: (5/13, 12/13). Step 3: Final Answer: The unit vector is <5/13, 12/13>. Final Answer:
Unit Vector (Normalization)
Given vectors u = <3, 2> and v = <-1, 4>, find u + v.
Step 1: Add the first components: 3 + (-1) = 2. Step 2: Add the second components: 2 + 4 = 6. Step 3: Final Answer: u + v = <2, 6>. Final Answer:
Vector Addition