How to Find Vector Components?

Components of a vector help to split a given vector into parts according to different directions. Let us talk more about the components of a vector and how to find the components of a vector.

How to Find Vector Components?

The components of a vector give a split of the vector. The vector is split according to each of the axes and we can calculate the components of a vector.

Related Topics

Step by step guide to vector components

  • Any vector in a two-dimensional coordinate system can be broken down into its \(x\) and \(y\)-components.

\(v⃗ =(v_x,v_y)\)

For example, in the picture given below, the vector \(v⃗\) is divided into two components, \(v_x\) and \(v_y\). Let the angle between the vector and its \(x\)-component be \(θ\).

In the diagram below, the vector and its component form a right-angled triangle:

In the above figure, the components can be easily and quickly read. The vector in the component form is \(v⃗ =(4,5)\).

  • Trigonometric ratios show the relationship between the magnitude of the vector and the components of the vector.

\(\color{blue}{cos θ=\frac{v_x}{v}}\) → \(\color{blue}{v_x= v \cos θ}\)

\(\color{blue}{sin θ=\frac{v_y}{v}}\) → \(\color{blue}{v_y= v \sin θ}\)

  • Using “Pythagoras theorem” in right triangles with lengths \(v_x\) and \(v_y\):

\(\color{blue}{|v|=\sqrt{v_x^2+v_y^2}}\)

Note1:

Find the magnitude and direction of the vector with respect to the components of a vector. In this case, use the following formulas:

The magnitude of the vector is \(\color{blue}{|v|=\sqrt{v_x^2+v_y^2}}\).

To find the direction of the vector, solve \(\color{blue}{tan θ=\frac{v_y}{v_x}}\) for \(θ\).

Note 2:

Find the components of a vector according to the magnitude and direction of a vector. In this case, use the following formulas:

\(\color{blue}{v_x= v \cos θ}\)

\(\color{blue}{v_y= v \sin θ}\)

Vector Components – Example 1:

The magnitude of a vector \(v⃗\) is \(20\) units and the direction of the vector is \(60°\) with the horizontal. Find the components of the vector.

To find the components of a vector use these formulas:

\(\color{blue}{v_x= v \cos θ}\)

\(\color{blue}{v_y= v \sin θ}\)

\(v_x= v\cos 60° \) → \(v_x= 20×\frac{1}{2}= \frac{20}{2}=10\)

\(v_y= v\sin 60° \) → \(v_y= 20×\frac{\sqrt{3}}{2}= \frac{20\sqrt{3}}{2} =10\sqrt{3}\)

So, the vector \(v⃗\) is \((10, 10\sqrt{3})\).

Vector Components – Example 2:

Find the \(x\) and \(y\) components of a vector having a magnitude of \(10\) and make an angle of \(45\) degrees with the positive \(x\)-axis.

To find the components of a vector use these formulas:

\(\color{blue}{v_x= v \cos θ}\)

\(\color{blue}{v_y= v \sin θ}\)

\(v_x= v\cos 45° \) → \(v_x= 10×\frac{\sqrt{2}}{2}= \frac{10\sqrt{2}}{2} =5\sqrt{2}\)

\(v_y= v\sin 45° \) → \(v_y= 10×\frac{\sqrt{2}}{2}= \frac{10\sqrt{2}}{2} =5\sqrt{2}\)

So, the \(x\)-component and the \(y\)-components of the vector are both equal to \(5\sqrt{2}\).

Exercises for Vector Components

  1. Find the value of \( θ \), if \(v_x=15\) and \(v_y=8.66\).
  2. Find out the magnitude of a vector \(OA=(-3,4)\).
  3. Find the components of the vector, if the magnitude of a vector \(v⃗\) is \(6\) units and the direction of the vector is \(30°\) with the horizontal.
  4. Find the direction of \((-4,3)\).
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{θ=30^\circ}\)
  2. \(\color{blue}{|OA|=5}\)
  3. \(\color{blue}{v⃗=3, 3\sqrt{3}}\)
  4. \(\color{blue}{θ=143.13^\circ}\)

Related to "How to Find Vector Components?"

How to Fall in Love with Math and Studying?How to Fall in Love with Math and Studying?
How to Apply Trigonometry to General Triangles?How to Apply Trigonometry to General Triangles?
How to Determine Limits Using the Squeeze Theorem?How to Determine Limits Using the Squeeze Theorem?
How to Determine Limits Using Algebraic Manipulation?How to Determine Limits Using Algebraic Manipulation?
How to Estimate Limit Values from the Graph?How to Estimate Limit Values from the Graph?
Properties of LimitsProperties of Limits
How to Find the Expected Value of a Random Variable?How to Find the Expected Value of a Random Variable?
How to Define Limits Analytically Using Correct Notation?How to Define Limits Analytically Using Correct Notation?
How to Solve Multiplication Rule for Probabilities?How to Solve Multiplication Rule for Probabilities?
How to Solve Venn Diagrams and the Addition Rule?How to Solve Venn Diagrams and the Addition Rule?

What people say about "How to Find Vector Components?"?

No one replied yet.

Leave a Reply