Textbook Question
Write each vector in the form a i + b j.
〈2, 0〉
665
views
8. Vectors
Geometric Vectors
Problem 7.41b
Textbook Question
Given vectors u and v, find: 2u + 3v.
u = 2i, v = i + j
Verified step by step guidance1
Identify the components of vector \( \mathbf{u} \) and vector \( \mathbf{v} \). Here, \( \mathbf{u} = 2\mathbf{i} \) and \( \mathbf{v} = \mathbf{i} + \mathbf{j} \).
Multiply vector \( \mathbf{u} \) by 2: \( 2\mathbf{u} = 2(2\mathbf{i}) = 4\mathbf{i} \).
Multiply vector \( \mathbf{v} \) by 3: \( 3\mathbf{v} = 3(\mathbf{i} + \mathbf{j}) = 3\mathbf{i} + 3\mathbf{j} \).
Add the results from the previous two steps: \( 2\mathbf{u} + 3\mathbf{v} = 4\mathbf{i} + (3\mathbf{i} + 3\mathbf{j}) \).
Combine like terms to simplify: \( 4\mathbf{i} + 3\mathbf{i} + 3\mathbf{j} = (4 + 3)\mathbf{i} + 3\mathbf{j} \).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vector Addition
Vector addition involves combining two or more vectors to produce a resultant vector. This is done by adding their corresponding components. For example, if vector u has components (u1, u2) and vector v has components (v1, v2), the resultant vector w is given by w = (u1 + v1, u2 + v2). Understanding this concept is crucial for solving problems involving multiple vectors.
Recommended video:
05:29
Adding Vectors Geometrically
Scalar Multiplication
Scalar multiplication refers to the process of multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction. For instance, if vector u = (u1, u2) is multiplied by a scalar k, the resulting vector is ku = (ku1, ku2). This concept is essential for manipulating vectors in expressions like 2u and 3v in the given problem.
Recommended video:
05:05Multiplying Vectors By Scalars
Unit Vectors
Unit vectors are vectors with a magnitude of one, often used to indicate direction. In the context of the problem, the vectors u and v are expressed in terms of the standard unit vectors i and j, which represent the x and y directions, respectively. Understanding unit vectors helps in visualizing and performing operations on vectors in a coordinate system.
Recommended video:
04:04Unit Vector in the Direction of a Given Vector
3:48mWatch next
Master Introduction to Vectors with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice