Vector Addition and Scalar Multiplication

Oct 13, 2025 by windailys.com 95 views

Vector Operations: Finding $4\vec{a} - 3\vec{b}$

In this article, we will explore the concept of vector operations, specifically the process of finding the result of a linear combination of two vectors. We will use the given vectors $\vec{a} = \langle -1, -2 \rangle$ and $\vec{b} = \langle 2, -1 \rangle$ to find the result of $4\vec{a} - 3\vec{b}$ .

A vector is a mathematical object that has both magnitude (length) and direction. It can be represented graphically as an arrow in a coordinate system. Vectors can be added and scaled using various operations, which are essential in many areas of mathematics and physics.

Vector Addition and Scalar Multiplication

Vector addition is the process of combining two or more vectors to form a new vector. The resulting vector is the sum of the individual vectors. Scalar multiplication is the process of multiplying a vector by a scalar (a number) to obtain a new vector.

To find $4\vec{a} - 3\vec{b}$ , we need to perform two operations:

Multiply $\vec{a}$ by 4: $4\vec{a} = 4 \langle -1, -2 \rangle = \langle -4, -8 \rangle$
Multiply $\vec{b}$ by 3: $3\vec{b} = 3 \langle 2, -1 \rangle = \langle 6, -3 \rangle$
Subtract $3\vec{b}$ from $4\vec{a}$ : $4\vec{a} - 3\vec{b} = \langle -4, -8 \rangle - \langle 6, -3 \rangle = \langle -4 - 6, -8 - (-3) \rangle = \langle -10, -5 \rangle$

In this article, we have demonstrated how to find the result of $4\vec{a} - 3\vec{b}$ using the given vectors $\vec{a} = \langle -1, -2 \rangle$ and $\vec{b} = \langle 2, -1 \rangle$ . We have performed scalar multiplication and vector subtraction to obtain the final result.

The correct answer is:

$\boxed{\langle -10, -5 \rangle}$

This problem is a basic example of vector operations, which are essential in many areas of mathematics and physics. Understanding vector operations is crucial for solving problems in linear algebra, calculus, and physics.

Find $2\vec{a} + 3\vec{b}$ using the given vectors $\vec{a} = \langle -1, -2 \rangle$ and $\vec{b} = \langle 2, -1 \rangle$ .
Find $3\vec{a} - 2\vec{b}$ using the given vectors $\vec{a} = \langle -1, -2 \rangle$ and $\vec{b} = \langle 2, -1 \rangle$ .

$2\vec{a} + 3\vec{b} = 2 \langle -1, -2 \rangle + 3 \langle 2, -1 \rangle = \langle -2, -4 \rangle + \langle 6, -3 \rangle = \langle -2 + 6, -4 - 3 \rangle = \langle 4, -7 \rangle$
$3\vec{a} - 2\vec{b} = 3 \langle -1, -2 \rangle - 2 \langle 2, -1 \rangle = \langle -3, -6 \rangle - \langle 4, -2 \rangle = \langle -3 - 4, -6 - (-2) \rangle = \langle -7, -4 \rangle$
Vector Operations: Q&A =========================

In our previous article, we explored the concept of vector operations, specifically the process of finding the result of a linear combination of two vectors. We used the given vectors $\vec{a} = \langle -1, -2 \rangle$ and $\vec{b} = \langle 2, -1 \rangle$ to find the result of $4\vec{a} - 3\vec{b}$ . In this article, we will answer some frequently asked questions related to vector operations.

Q1: What is the difference between vector addition and scalar multiplication?

A1: Vector addition is the process of combining two or more vectors to form a new vector. The resulting vector is the sum of the individual vectors. Scalar multiplication is the process of multiplying a vector by a scalar (a number) to obtain a new vector.

Q2: How do I find the result of $2\vec{a} + 3\vec{b}$ using the given vectors $\vec{a} = \langle -1, -2 \rangle$ and $\vec{b} = \langle 2, -1 \rangle$ ?

A2: To find $2\vec{a} + 3\vec{b}$ , you need to perform two operations:

Multiply $\vec{a}$ by 2: $2\vec{a} = 2 \langle -1, -2 \rangle = \langle -2, -4 \rangle$
Multiply $\vec{b}$ by 3: $3\vec{b} = 3 \langle 2, -1 \rangle = \langle 6, -3 \rangle$
Add $2\vec{a}$ and $3\vec{b}$ : $2\vec{a} + 3\vec{b} = \langle -2, -4 \rangle + \langle 6, -3 \rangle = \langle -2 + 6, -4 - 3 \rangle = \langle 4, -7 \rangle$

Q3: How do I find the result of $3\vec{a} - 2\vec{b}$ using the given vectors $\vec{a} = \langle -1, -2 \rangle$ and $\vec{b} = \langle 2, -1 \rangle$ ?

A3: To find $3\vec{a} - 2\vec{b}$ , you need to perform two operations:

Multiply $\vec{a}$ by 3: $3\vec{a} = 3 \langle -1, -2 \rangle = \langle -3, -6 \rangle$
Multiply $\vec{b}$ by 2: $2\vec{b} = 2 \langle 2, -1 \rangle = \langle 4, -2 \rangle$
Subtract $2\vec{b}$ from $3\vec{a}$ : $3\vec{a} - 2\vec{b} = \langle -3, -6 \rangle - \langle 4, -2 \rangle = \langle -3 - 4, -6 - (-2) \rangle = \langle -7, -4 \rangle$

Q4: What is the formula for finding the result of $c\vec{a} + d\vec{b}$ using the given vectors $\vec{a} = \langle x_1, y_1 \rangle$ and $\vec{b} = \langle x_2, y_2 \rangle$ ?

A4: The formula for finding the result of $c\vec{a} + d\vec{b}$ is:

$c\vec{a} + d\vec{b} = \langle cx_1 + dx_2, cy_1 + dy_2 \rangle$

Q5: How do I find the magnitude of a vector $\vec{v} = \langle x, y \rangle$ ?

A5: The magnitude of a vector $\vec{v} = \langle x, y \rangle$ is given by the formula:

$|\vec{v}| = \sqrt{x^2 + y^2}$

In this article, we have answered some frequently asked questions related to vector operations. We have provided step-by-step solutions to various problems and formulas for finding the result of linear combinations of vectors. Understanding vector operations is crucial for solving problems in linear algebra, calculus, and physics.

Find $2\vec{a} + 3\vec{b}$ using the given vectors $\vec{a} = \langle -1, -2 \rangle$ and $\vec{b} = \langle 2, -1 \rangle$ .
Find $3\vec{a} - 2\vec{b}$ using the given vectors $\vec{a} = \langle -1, -2 \rangle$ and $\vec{b} = \langle 2, -1 \rangle$ .
Find the magnitude of a vector $\vec{v} = \langle 3, 4 \rangle$ .

$2\vec{a} + 3\vec{b} = \langle 4, -7 \rangle$
$3\vec{a} - 2\vec{b} = \langle -7, -4 \rangle$
$|\vec{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$