Calculate The Following Expression:${ \frac{3}{2} \cdot \left(\begin{array}{c} 1 \ 5 \ 3 \end{array}\right) }$

Mar 11, 2025 by ADMIN 112 views

**Understanding Vector Multiplication in Mathematics**

Introduction

In mathematics, vectors are an essential concept used to represent quantities with both magnitude and direction. Vector multiplication is a fundamental operation that combines two or more vectors to produce a new vector. In this article, we will explore the process of calculating the expression $\frac{3}{2} \cdot \left(\begin{array}{c} 1 \\ 5 \\ 3 \end{array}\right)$ , which involves multiplying a scalar by a vector.

What is Vector Multiplication?

Vector multiplication is a binary operation that combines two vectors to produce a new vector. There are two types of vector multiplication: scalar multiplication and dot product. Scalar multiplication involves multiplying a vector by a scalar (a number), while the dot product involves multiplying two vectors to produce a scalar value.

Scalar Multiplication

Scalar multiplication is a simple operation that involves multiplying each component of a vector by a scalar. The resulting vector has the same direction as the original vector, but its magnitude is scaled by the scalar value.

Calculating the Expression

To calculate the expression $\frac{3}{2} \cdot \left(\begin{array}{c} 1 \\ 5 \\ 3 \end{array}\right)$ , we need to multiply each component of the vector by the scalar value $\frac{3}{2}$ .

$\frac{3}{2} \cdot \left(\begin{array}{c} 1 \\ 5 \\ 3 \end{array}\right) = \left(\begin{array}{c} \frac{3}{2} \cdot 1 \\ \frac{3}{2} \cdot 5 \\ \frac{3}{2} \cdot 3 \end{array}\right)$

Performing the Multiplication

Now, let's perform the multiplication for each component of the vector.

$\frac{3}{2} \cdot 1 = \frac{3}{2}$

$\frac{3}{2} \cdot 5 = \frac{15}{2}$

$\frac{3}{2} \cdot 3 = \frac{9}{2}$

The Resulting Vector

The resulting vector is:

$\left(\begin{array}{c} \frac{3}{2} \\ \frac{15}{2} \\ \frac{9}{2} \end{array}\right)$

Conclusion

In this article, we explored the process of calculating the expression $\frac{3}{2} \cdot \left(\begin{array}{c} 1 \\ 5 \\ 3 \end{array}\right)$ , which involves multiplying a scalar by a vector. We performed the scalar multiplication for each component of the vector and obtained the resulting vector.

Real-World Applications

Vector multiplication has numerous real-world applications in physics, engineering, and computer science. Some examples include:

Physics: Vector multiplication is used to describe the motion of objects in three-dimensional space. For example, the velocity of an object can be represented as a vector, and its magnitude and direction can be calculated using vector multiplication.
Engineering: Vector multiplication is used in computer-aided design (CAD) software to create 3D models of objects. The software uses vector multiplication to calculate the position and orientation of objects in 3D space.
Computer Science: Vector multiplication is used in computer graphics to create 3D models and animations. The software uses vector multiplication to calculate the position and orientation of objects in 3D space.

Common Mistakes

When performing vector multiplication, it's essential to remember the following common mistakes:

Incorrect scalar value: Make sure to use the correct scalar value when performing scalar multiplication.
Incorrect vector component: Make sure to use the correct vector component when performing scalar multiplication.
Incorrect order of operations: Make sure to perform the operations in the correct order when performing scalar multiplication.

Best Practices

When performing vector multiplication, follow these best practices:

Use the correct scalar value: Use the correct scalar value when performing scalar multiplication.
Use the correct vector component: Use the correct vector component when performing scalar multiplication.
Perform operations in the correct order: Perform the operations in the correct order when performing scalar multiplication.

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about vector multiplication.

Q: What is vector multiplication?

A: Vector multiplication is a binary operation that combines two vectors to produce a new vector. There are two types of vector multiplication: scalar multiplication and dot product.

Q: What is scalar multiplication?

A: Scalar multiplication involves multiplying a vector by a scalar (a number). The resulting vector has the same direction as the original vector, but its magnitude is scaled by the scalar value.

Q: What is the dot product?

A: The dot product involves multiplying two vectors to produce a scalar value. It is also known as the inner product or scalar product.

Q: How do I perform scalar multiplication?

A: To perform scalar multiplication, you need to multiply each component of the vector by the scalar value. For example, if you have a vector $\left(\begin{array}{c} 1 \\ 5 \\ 3 \end{array}\right)$ and a scalar value of $2$ , the resulting vector would be $\left(\begin{array}{c} 2 \cdot 1 \\ 2 \cdot 5 \\ 2 \cdot 3 \end{array}\right) = \left(\begin{array}{c} 2 \\ 10 \\ 6 \end{array}\right)$ .

Q: How do I perform the dot product?

A: To perform the dot product, you need to multiply the corresponding components of the two vectors and add them together. For example, if you have two vectors $\left(\begin{array}{c} 1 \\ 5 \\ 3 \end{array}\right)$ and $\left(\begin{array}{c} 2 \\ 4 \\ 6 \end{array}\right)$ , the dot product would be $(1 \cdot 2) + (5 \cdot 4) + (3 \cdot 6) = 2 + 20 + 18 = 40$ .

Q: What are some common mistakes to avoid when performing vector multiplication?

A: Some common mistakes to avoid when performing vector multiplication include:

Incorrect scalar value: Make sure to use the correct scalar value when performing scalar multiplication.
Incorrect vector component: Make sure to use the correct vector component when performing scalar multiplication.
Incorrect order of operations: Make sure to perform the operations in the correct order when performing scalar multiplication.

Q: What are some real-world applications of vector multiplication?

A: Vector multiplication has numerous real-world applications in physics, engineering, and computer science. Some examples include:

Physics: Vector multiplication is used to describe the motion of objects in three-dimensional space. For example, the velocity of an object can be represented as a vector, and its magnitude and direction can be calculated using vector multiplication.
Engineering: Vector multiplication is used in computer-aided design (CAD) software to create 3D models of objects. The software uses vector multiplication to calculate the position and orientation of objects in 3D space.
Computer Science: Vector multiplication is used in computer graphics to create 3D models and animations. The software uses vector multiplication to calculate the position and orientation of objects in 3D space.

Q: How do I choose the correct vector multiplication operation?

A: To choose the correct vector multiplication operation, you need to consider the context of the problem. If you are working with a scalar value and a vector, you should use scalar multiplication. If you are working with two vectors, you should use the dot product.

Q: What are some best practices for performing vector multiplication?

A: Some best practices for performing vector multiplication include:

Use the correct scalar value: Use the correct scalar value when performing scalar multiplication.
Use the correct vector component: Use the correct vector component when performing scalar multiplication.
Perform operations in the correct order: Perform the operations in the correct order when performing scalar multiplication.

Conclusion

In conclusion, vector multiplication is a fundamental operation in mathematics that combines two or more vectors to produce a new vector. By understanding the different types of vector multiplication, including scalar multiplication and the dot product, you can perform vector multiplication with confidence. Remember to avoid common mistakes, follow best practices, and choose the correct vector multiplication operation for your problem.