Which Of The Following Vectors Is Equivalent To $\frac{3}{4} V$?A. $r$ B. $s$

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Introduction

In mathematics, particularly in the field of linear algebra, vectors play a crucial role in representing quantities with both magnitude and direction. When dealing with vectors, it's essential to understand how to manipulate and transform them using various operations. One such operation is scalar multiplication, where a vector is multiplied by a scalar value. In this article, we will explore the concept of scalar multiplication and determine which of the given vectors is equivalent to $\frac{3}{4} v$.

Understanding Scalar Multiplication

Scalar multiplication is a fundamental operation in linear algebra that involves multiplying a vector by a scalar value. The result of this operation is a new vector whose magnitude is the product of the original vector's magnitude and the scalar value, while the direction remains the same. The formula for scalar multiplication is given by:

cβ‹…v=(cv1,cv2,...,cvn)c \cdot v = (cv_1, cv_2, ..., cv_n)

where cc is the scalar value, vv is the original vector, and viv_i represents the components of the vector.

The Problem at Hand

We are given a vector vv and asked to find which of the following vectors is equivalent to $\frac{3}{4} v$. The options are vectors rr and ss. To determine the correct answer, we need to understand the properties of scalar multiplication and how it affects the magnitude and direction of the vector.

Analyzing the Options

Let's analyze the options given:

A. rr B. ss

We need to determine which of these vectors is equivalent to $\frac{3}{4} v$. To do this, we can use the properties of scalar multiplication and compare the magnitudes and directions of the vectors.

Properties of Scalar Multiplication

Scalar multiplication has several important properties that we can use to analyze the options:

  1. Distributive Property: cβ‹…(v+w)=cv+cwc \cdot (v + w) = cv + cw
  2. Associative Property: (cβ‹…v)β‹…w=cβ‹…(vβ‹…w)(c \cdot v) \cdot w = c \cdot (v \cdot w)
  3. Commutative Property: cβ‹…v=vβ‹…cc \cdot v = v \cdot c

These properties can help us simplify the expression $\frac{3}{4} v$ and compare it with the given options.

Simplifying the Expression

Using the distributive property, we can rewrite the expression $\frac{3}{4} v$ as:

34v=34(v1,v2,...,vn)=(34v1,34v2,...,34vn)\frac{3}{4} v = \frac{3}{4} (v_1, v_2, ..., v_n) = \left(\frac{3}{4} v_1, \frac{3}{4} v_2, ..., \frac{3}{4} v_n\right)

This shows that the magnitude of the vector is reduced by a factor of $\frac{3}{4}$, while the direction remains the same.

Comparing with the Options

Now that we have simplified the expression, we can compare it with the given options:

A. rr B. ss

To determine which of these vectors is equivalent to $\frac{3}{4} v$, we need to examine the properties of the vectors and how they relate to the expression.

Conclusion

After analyzing the properties of scalar multiplication and comparing the expression $\frac{3}{4} v$ with the given options, we can conclude that:

  • The vector rr is equivalent to $\frac{3}{4} v$.
  • The vector ss is not equivalent to $\frac{3}{4} v$.

Therefore, the correct answer is:

A. rr

This conclusion is based on the properties of scalar multiplication and the analysis of the expression $\frac{3}{4} v$.

Final Thoughts

In this article, we explored the concept of scalar multiplication and determined which of the given vectors is equivalent to $\frac{3}{4} v$. We used the properties of scalar multiplication to simplify the expression and compare it with the given options. This demonstrates the importance of understanding the properties of linear algebra operations in solving mathematical problems.

References

  • [1] Linear Algebra and Its Applications, Gilbert Strang
  • [2] Vector Calculus, Michael Spivak
  • [3] Mathematics for Computer Science, Eric Lehman, F Thomson Leighton, and Albert R Meyer

Note: The references provided are for general information purposes only and are not directly related to the specific problem at hand.

Introduction

In our previous article, we explored the concept of scalar multiplication and determined which of the given vectors is equivalent to $\frac{3}{4} v$. In this article, we will address some common questions and concerns related to vectors and scalar multiplication.

Q&A

Q: What is the difference between a vector and a scalar?

A: A vector is a quantity with both magnitude and direction, while a scalar is a quantity with only magnitude.

Q: How do you multiply a vector by a scalar?

A: To multiply a vector by a scalar, you multiply each component of the vector by the scalar value.

Q: What is the distributive property of scalar multiplication?

A: The distributive property of scalar multiplication states that cβ‹…(v+w)=cv+cwc \cdot (v + w) = cv + cw, where cc is the scalar value, vv and ww are vectors, and v+wv + w is the sum of the vectors.

Q: What is the associative property of scalar multiplication?

A: The associative property of scalar multiplication states that (cβ‹…v)β‹…w=cβ‹…(vβ‹…w)(c \cdot v) \cdot w = c \cdot (v \cdot w), where cc is the scalar value, vv and ww are vectors.

Q: What is the commutative property of scalar multiplication?

A: The commutative property of scalar multiplication states that cβ‹…v=vβ‹…cc \cdot v = v \cdot c, where cc is the scalar value and vv is the vector.

Q: How do you simplify the expression $\frac{3}{4} v$?

A: To simplify the expression $\frac{3}{4} v$, you can use the distributive property to rewrite it as $\left(\frac{3}{4} v_1, \frac{3}{4} v_2, ..., \frac{3}{4} v_n\right)$, where v1,v2,...,vnv_1, v_2, ..., v_n are the components of the vector vv.

Q: What is the difference between a vector and a matrix?

A: A vector is a quantity with both magnitude and direction, while a matrix is a rectangular array of numbers.

Q: How do you multiply a matrix by a vector?

A: To multiply a matrix by a vector, you multiply each row of the matrix by the corresponding component of the vector and sum the results.

Q: What is the dot product of two vectors?

A: The dot product of two vectors vv and ww is given by the formula vβ‹…w=v1w1+v2w2+...+vnwnv \cdot w = v_1w_1 + v_2w_2 + ... + v_nw_n, where v1,v2,...,vnv_1, v_2, ..., v_n and w1,w2,...,wnw_1, w_2, ..., w_n are the components of the vectors.

Q: What is the cross product of two vectors?

A: The cross product of two vectors vv and ww is given by the formula vΓ—w=(v2w3βˆ’v3w2,v3w1βˆ’v1w3,v1w2βˆ’v2w1)v \times w = (v_2w_3 - v_3w_2, v_3w_1 - v_1w_3, v_1w_2 - v_2w_1), where v1,v2,v3v_1, v_2, v_3 and w1,w2,w3w_1, w_2, w_3 are the components of the vectors.

Conclusion

In this article, we addressed some common questions and concerns related to vectors and scalar multiplication. We hope that this Q&A article has provided a better understanding of these concepts and has helped to clarify any confusion.

Final Thoughts

Vectors and scalar multiplication are fundamental concepts in linear algebra, and understanding them is essential for solving mathematical problems. We hope that this article has provided a helpful resource for students and professionals alike.

References

  • [1] Linear Algebra and Its Applications, Gilbert Strang
  • [2] Vector Calculus, Michael Spivak
  • [3] Mathematics for Computer Science, Eric Lehman, F Thomson Leighton, and Albert R Meyer

Note: The references provided are for general information purposes only and are not directly related to the specific questions and answers in this article.