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Chapter 5: Problem 27

Perform the indicated operations where \(u=3 i-2 j\) and \(v=-2 i+3 j\). $$6 u+2 v$$

Short Answer

Expert verified
The resultant vector after performing the indicated operations is \(14i-6j\).

Step by step solution

01

Perform Scalar Multiplication on Both Vectors

The first step involves performing scalar multiplication on both vectors. Multiply each vector by the scalar assigned to them in the operation. This implies, \(6u = 6 * (3i-2j) = 18i-12j\) and \(2v = 2 * (-2i+3j) = -4i+6j.\)
02

Perform Vector Addition

The second step is the addition of the two multiplied vectors which were obtained in the first step. This means, \(6u+2v = (18i-12j) + (-4i+6j) = 14i-6j\)
03

Present the Final Result

The final result after performing the scalar multiplication and vector addition operations is \(14i-6j\), which is the required vector.

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Key Concepts

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

Scalar Multiplication
Scalar multiplication is a basic operation in vector mathematics that involves multiplying a vector by a scalar (a real number). This process scales the vector by the given factor without changing its direction, unless the scalar is negative, which would also reverse the direction of the vector.
When you multiply a vector by a scalar:
  • You multiply each component of the vector by the scalar.
  • The vector's magnitude changes but not its direction unless the scalar is negative.
For the vector \( u = 3i - 2j \) and scalar 6, multiplying gives:
  • \( 6u = 6(3i - 2j) = 18i - 12j \)
Similarly, for the vector \( v = -2i + 3j \) and scalar 2, we have:
  • \( 2v = 2(-2i + 3j) = -4i + 6j \)
Vector Addition
Vector addition is the process of combining two or more vectors to produce a new vector. Imagine moving along one vector, then starting from its endpoint and moving along the second vector; the combined movement equals the resultant vector.
Here is how vector addition works:
  • Sum the corresponding components of the vectors.
  • The components are typically the horizontal and vertical parts of each vector.
From the exercise, after performing scalar multiplication, we have two vectors: \( 18i - 12j \) and \( -4i + 6j \). Adding these vectors involves combining their components:
  • Horizontal components: \( 18i + (-4i) = 14i \)
  • Vertical components: \( -12j + 6j = -6j \)
Thus, the resultant vector from adding these two vectors is \( 14i - 6j \).
i and j Unit Vectors
In vector mathematics, unit vectors are vectors with a magnitude of 1. They are used as building blocks for other vectors and allow for convenient mathematical expressions and operations. The two most common unit vectors in two-dimensional space are:
  • \( i \): represents a unit vector along the x-axis (horizontal direction).
  • \( j \): represents a unit vector along the y-axis (vertical direction).
When a vector is expressed as \( ai + bj \), it means the vector has a horizontal component, \( a \), times the unit vector \( i \), and a vertical component, \( b \), times the unit vector \( j \).
For instance, in our exercise:
  • The vector \( u = 3i - 2j \) means it points 3 units along the x-axis and -2 units along the y-axis.
  • The vector \( v = -2i + 3j \) means it points -2 units along the x-axis and 3 units along the y-axis.
Understanding \( i \) and \( j \) is crucial for visualizing and performing operations on vectors.

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Most popular questions from this chapter

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