Given That $b = \binom{2}{4}$, $c = \binom{3}{2}$, And $ A = 3 C − 2 B A = 3c - 2b A = 3 C − 2 B [/tex], Find The Magnitude Of $a$, Correct To 2 Decimal Places.

Introduction

In mathematics, vectors are used to represent quantities with both magnitude and direction. The magnitude of a vector is a measure of its size or length. In this article, we will discuss how to calculate the magnitude of a vector given its components. We will use the binomial coefficient to find the magnitude of a vector.

Understanding Binomial Coefficients

A binomial coefficient, often referred to as "n choose k", is a mathematical expression that represents the number of ways to choose k items from a set of n items without regard to the order of selection. It is denoted by the symbol $\binom{n}{k}$ and is calculated as:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

where $n!$ represents the factorial of n, which is the product of all positive integers from 1 to n.

Calculating the Components of Vector a

Given that $b = \binom{2}{4}$, $c = \binom{3}{2}$, and $a = 3c - 2b$, we need to calculate the values of b and c before we can find the value of a.

First, let's calculate the value of b:

$$b = \binom{2}{4} = \frac{2!}{4!(2-4)!} = \frac{2}{24} = \frac{1}{12}$$

Next, let's calculate the value of c:

$$c = \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{6}{2 \cdot 1} = 3$$

Now that we have the values of b and c, we can calculate the value of a:

$$a = 3c - 2b = 3(3) - 2\left(\frac{1}{12}\right) = 9 - \frac{1}{6} = \frac{53}{6}$$

Calculating the Magnitude of Vector a

The magnitude of a vector is calculated using the formula:

$$|a| = \sqrt{a_1^2 + a_2^2 + ... + a_n^2}$$

where $a_1, a_2, ..., a_n$ are the components of the vector.

In this case, we have a single component, so the magnitude of vector a is:

$$|a| = \sqrt{\left(\frac{53}{6}\right)^2} = \sqrt{\frac{2809}{36}} = \frac{\sqrt{2809}}{6}$$

Evaluating the Magnitude of Vector a

To find the magnitude of vector a, we need to evaluate the expression $\frac{\sqrt{2809}}{6}$.

Using a calculator, we get:

$$\frac{\sqrt{2809}}{6} \approx 6.69$$

Therefore, the magnitude of vector a is approximately 6.69, correct to 2 decimal places.

Conclusion

Q: What is the binomial coefficient?

A: The binomial coefficient, often referred to as "n choose k", is a mathematical expression that represents the number of ways to choose k items from a set of n items without regard to the order of selection. It is denoted by the symbol $\binom{n}{k}$ and is calculated as:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

Q: How do I calculate the binomial coefficient?

A: To calculate the binomial coefficient, you need to use the formula:

$$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$

where $n!$ represents the factorial of n, which is the product of all positive integers from 1 to n.

Q: What is the magnitude of a vector?

A: The magnitude of a vector is a measure of its size or length. It is calculated using the formula:

$$|a| = \sqrt{a_1^2 + a_2^2 + ... + a_n^2}$$

where $a_1, a_2, ..., a_n$ are the components of the vector.

Q: How do I calculate the magnitude of a vector?

A: To calculate the magnitude of a vector, you need to use the formula:

$$|a| = \sqrt{a_1^2 + a_2^2 + ... + a_n^2}$$

where $a_1, a_2, ..., a_n$ are the components of the vector.

Q: What is the difference between the magnitude and the length of a vector?

A: The magnitude and the length of a vector are the same thing. They are both measures of the size or length of a vector.

Q: Can I use the magnitude of a vector to find the length of a line segment?

A: Yes, you can use the magnitude of a vector to find the length of a line segment. If you have a vector that represents the direction and magnitude of a line segment, you can use the magnitude of the vector to find the length of the line segment.

Q: How do I find the magnitude of a vector with multiple components?

A: To find the magnitude of a vector with multiple components, you need to use the formula:

$$|a| = \sqrt{a_1^2 + a_2^2 + ... + a_n^2}$$

where $a_1, a_2, ..., a_n$ are the components of the vector.

Q: Can I use a calculator to find the magnitude of a vector?

A: Yes, you can use a calculator to find the magnitude of a vector. Simply enter the components of the vector into the calculator and use the formula:

$$|a| = \sqrt{a_1^2 + a_2^2 + ... + a_n^2}$$

to find the magnitude of the vector.

Q: What is the significance of the magnitude of a vector in real-world applications?

A: The magnitude of a vector is significant in many real-world applications, including physics, engineering, and computer science. It is used to represent the size or length of a vector, which is essential in many calculations and applications.

Conclusion

In this article, we have answered some frequently asked questions about calculating the magnitude of a vector. We have discussed the binomial coefficient, the magnitude of a vector, and how to calculate the magnitude of a vector with multiple components. We have also provided examples and explanations to help you understand the concepts.