Component Form of a Vector

Oct 13, 2025 by miodeya.com 137 views

Introduction

In mathematics, vectors are used to represent quantities with both magnitude and direction. The component form of a vector is a way to express it as an ordered pair of numbers, which can be used to perform various calculations. In this article, we will discuss how to find the component form of a vector given two points in a coordinate plane.

What is a Vector?

A vector is a mathematical object that has both magnitude (length) and direction. It can be represented graphically as an arrow in a coordinate plane, with the tail of the arrow at the initial point and the head at the terminal point. Vectors can be added, subtracted, and multiplied, making them a fundamental concept in mathematics and physics.

Component Form of a Vector

The component form of a vector is an ordered pair of numbers that represents the vector. It is denoted by the symbol $\langle a, b \rangle$ , where $a$ is the x-component and $b$ is the y-component. The component form of a vector can be used to perform various calculations, such as finding the magnitude and direction of the vector.

Finding the Component Form of a Vector

To find the component form of a vector given two points in a coordinate plane, we need to follow these steps:

Identify the initial and terminal points: The initial point is the point from which the vector starts, and the terminal point is the point to which the vector ends.
Find the difference in x-coordinates: Subtract the x-coordinate of the initial point from the x-coordinate of the terminal point to find the x-component of the vector.
Find the difference in y-coordinates: Subtract the y-coordinate of the initial point from the y-coordinate of the terminal point to find the y-component of the vector.

Example Problem

Given the points $P(2,-6)$ and $R(8,3)$ , what is the component form of $\overrightarrow{PR}$ ?

Step 1: Identify the initial and terminal points

The initial point is $P(2,-6)$ , and the terminal point is $R(8,3)$ .

Step 2: Find the difference in x-coordinates

Subtract the x-coordinate of the initial point from the x-coordinate of the terminal point:

8 - 2 = 6

Step 3: Find the difference in y-coordinates

Subtract the y-coordinate of the initial point from the y-coordinate of the terminal point:

3 - (-6) = 3 + 6 = 9

Step 4: Write the component form of the vector

The component form of $\overrightarrow{PR}$ is $\langle 6, 9 \rangle$ .

Conclusion

In this article, we discussed how to find the component form of a vector given two points in a coordinate plane. We followed the steps of identifying the initial and terminal points, finding the difference in x-coordinates, finding the difference in y-coordinates, and writing the component form of the vector. We applied this concept to an example problem and found the component form of $\overrightarrow{PR}$ to be $\langle 6, 9 \rangle$ .

Answer

The correct answer is $\boxed{\langle 6, 9 \rangle}$ .

References

Introduction

In our previous article, we discussed how to find the component form of a vector given two points in a coordinate plane. In this article, we will answer some frequently asked questions related to vector calculations.

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

A: A scalar is a number that has only magnitude, whereas a vector is a mathematical object that has both magnitude and direction.

Q: How do you add two vectors?

A: To add two vectors, you need to add their corresponding components. For example, if you have two vectors $\langle a, b \rangle$ and $\langle c, d \rangle$ , their sum is $\langle a + c, b + d \rangle$ .

Q: How do you subtract two vectors?

A: To subtract two vectors, you need to subtract their corresponding components. For example, if you have two vectors $\langle a, b \rangle$ and $\langle c, d \rangle$ , their difference is $\langle a - c, b - d \rangle$ .

Q: What is the magnitude of a vector?

A: The magnitude of a vector is its length or size. It can be calculated using the formula $\sqrt{a^2 + b^2}$ , where $a$ and $b$ are the components of the vector.

Q: What is the direction of a vector?

A: The direction of a vector is the angle it makes with the positive x-axis. It can be calculated using the formula $\tan^{-1}\left(\frac{b}{a}\right)$ , where $a$ and $b$ are the components of the vector.

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

A: To multiply a vector by a scalar, you need to multiply its components by the scalar. For example, if you have a vector $\langle a, b \rangle$ and a scalar $c$ , their product is $\langle ca, cb \rangle$ .

Q: What is the dot product of two vectors?

A: The dot product of two vectors is a scalar value that represents the amount of "similarity" between the two vectors. It can be calculated using the formula $a_1a_2 + b_1b_2$ , where $a_1, a_2, b_1, b_2$ are the components of the two vectors.

Q: What is the cross product of two vectors?

A: The cross product of two vectors is a vector that is perpendicular to both of the original vectors. It can be calculated using the formula $\langle a_1b_2 - a_2b_1, a_2b_1 - a_1b_2 \rangle$ , where $a_1, a_2, b_1, b_2$ are the components of the two vectors.

Conclusion

In this article, we answered some frequently asked questions related to vector calculations. We discussed the difference between a vector and a scalar, how to add and subtract vectors, how to calculate the magnitude and direction of a vector, how to multiply a vector by a scalar, and how to calculate the dot and cross products of two vectors.