Components of a Vector

Components of a Vector refer to the parts of a vector that show how it influences each axis in a coordinate system. Vectors, defined by their magnitude (size) and direction, can be better understood by breaking them down into these components. This process is common in two or three dimensions, helping to explain how the vector works along each axis.

In two dimensions, a vector is split into components along the x and y axes. In three dimensions, it is split into components along the x, y, and z axes. Breaking down a vector helps us understand its parts. These parts can later be added together to recreate the full vector.

The components are like the "building blocks" of the vector, showing how much of the vector points in each direction. In 2D, these components can be found using trigonometric functions:

• Cosine (cos⁡) gives the horizontal (x-axis) component.
• Sine (sin⁡) gives the vertical (y-axis) component.

In 3D, the components are often given directly as the vector’s values along the x, y, and z axes.

By combining these components, we can fully describe the vector's magnitude and direction, which makes them a powerful tool in mathematics, physics, and engineering.

X and Y Components of a Vector

In a flat, two-dimensional setup, we look at the angle the vector makes with the horizontal x-axis. Suppose in the vector V and the angle it makes with the x-axis θ. The parts of V along the x and y axes are written as Vx and Vy. We find these parts using these formulas:

• V_x = V × cos θ
• V_y = V × sin θ

To find the total size of the vector (magnitude), the formula is:

|V| = square root of (V_x² + V_y²)

In three-dimensional space, vectors are represented like this:

Vector A equals a times the unit vector i plus b times the unit vector j plus c times the unit vector k where, i, j, and k are the unit vectors for the x, y, and z axes. The components of vector A with respect to the x, y, and z axes are a, b, and c respectively.

Formula of Components of a Vector

For a two-dimensional vector \vec{a} its components can be expressed as:

• a_x = |a| × cosθ
• a_y = |a| × sinθ

Here, |a| represents the magnitude of vector \vec{a} and θ is the angle formed by the vector with the positive x-axis.

Formula for Components in 3D

For a three-dimensional vector \vec{a} the components along the x, y, and z axes are given by:

• a_x = |a| × cos α
• a_y = |a| × cos β
• a_z = |a| × cos γ

In this case, α, β and γ are the angles between the vector and the positive x, y, and z axes, respectively. The magnitude |a| is the overall length of the vector.

How to Find the Components of a Vector?

The vector \vec{A} is called the component form. The values a, b, and c are known as the scalar components of vector A. The expressions a_i, b_j, and c_k are called the vector components. Also, a, b, and c are referred to as rectangular components.

To find the magnitude of A, you use the formula: |A| = \sqrt{a^2+b^2+c^2} , which means taking the square root of the sum of the squares of its individual components.

Components of a Vector Along b Vector

Breaking down a vector along another vector involves understanding how much of the first vector points in the same direction as the second one. We can calculate this length using the following formula:

Projection of A onto B = (Dot Product of A and B) / Magnitude of B

Components of a Vector along Another Vector

To find the components of vector along another vector, follow the given below steps:

• Step 1: Identify the Vectors

Determine the vectors involved. Let's call them Vector A and Vector B.

• Step 2: Find the Dot Product

Calculate the dot product of Vector A and Vector B. The dot product is found by multiplying the corresponding components of the vectors and adding the results.

• Step 3: Determine the Magnitude of Vector B

Find the magnitude (length) of Vector B. This involves taking the square root of the sum of the squares of its components.

• Step 4: Calculate the Projection

Use the dot product and the magnitude of Vector B to calculate the projection of Vector A onto Vector B. The projection represents the component of Vector A in the direction of Vector B.

Projection of A onto B= (Dot Product of A and B) / Magnitude of B

• Step 5: Find Components Along B

The components of Vector A along Vector B are the projections calculated in the previous step. These components show how much of Vector A points in the same direction as Vector B.

Learn, Vector Projection Formula

Vector Addition with Components

Some of the common methods for vector addition

• Scalar Addition
• Graphical Method
• Algebraic Method

Scalar Addition

Scalar addition involves adding the magnitudes of vectors. If you have two vectors, let's say (a) and (b), you add their magnitudes to get a new vector. For example, if (|a| = 3) and (|b| = 4), the scalar sum would be (|a| + |b| = 7).

Graphical Method

The graphical method represents vectors as arrows. To add vectors graphically, you place the tail of the second vector at the head of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the second vector.

This is like walking from the starting point to the ending point of each vector in sequence. The length and direction of the resultant vector represent the sum of the original vectors.

Algebraic Method

The algebraic method involves breaking down vectors into their components. If you have vectors (a) and (b) with components (ax, ay) and (bx, by) respectively, the sum is obtained by adding corresponding components:

Resultant vector = (a_x + b_x, a_y + b_y)

For example, if (a = 2, 3) and (b = 1, 4), the sum (a + b) would be (2 + 1, 3 + 4) = (3, 7)

Types of Vector Components

• Rectangular Components
• Polar Components

Rectangular Components of a Vector

Rectangular components are along the coordinate axes, typically denoted as a_x, a_y, and a_z for a three-dimensional vector. These components are found using trigonometric functions based on the angles between the vector and the coordinate axes.

For a two-dimensional vector (a), the components are given by:

• a_x = |a| × cosθ
• a_y = |a| × sinθ

In three dimensions, the components are:

• a_x = |a| × cos α
• a_y = |a| × cos β
• a_z = |a| × cos γ

Here, |a| represents the magnitude of the vector, and θ, α, β and γ are angles with respect to the coordinate axes.

Polar Components of a Vector

Polar components describe a vector's magnitude and direction using polar coordinates. In two dimensions, a vector (a) is represented as (r, θ), where (r) is the magnitude and θ is the angle with the positive x-axis.

The rectangular components (a_x) and (a_y) can be found using:

• a_x = r · cosθ
• a_y = r · sinθ

Polar components provide an alternative way to describe vectors, especially when dealing with circular or radial motion.

Orthogonal Vectors

In coordinate geometry, orthogonal representation refers to parameters that are at right angles to each other. In a three-dimensional orthogonal system, there are three axes perpendicular to each other, representing the x, y, and z axis.

Unit vectors are vectors with a length of one unit. In this context:

• \hat{u} represents a unit vector,
• \vec{v} represents the vector,
• |\vec{v}| represents the magnitude of the vector.

In orthogonal systems, we have three unit vectors, one for each direction, denoted as \hat{i} for the x-axis, \hat{j} for the y-axis, and \hat{k} for the z-axis.

For example, the point X(1, 1, 1) can be expressed using mutually perpendicular axes as points A(1, 0, 0), B(0, 1, 0), and C(0, 0, 1) on the x, y, and z axes, respectively.

The magnitudes of the vectors \hat{i}, \hat{j}, and \hat{k} are 1 along the x, y, and z axes, respectively. Using unit vectors, any vector in a three-dimensional coordinate system can be represented.

Also, Check

• Scalars and Vectors
• Dot and Cross Product of Vectors
• Magnitude of a Vector Formula

Solved Examples on Components of a Vector

Example 1: Consider a vector v with a magnitude of 5 units and an angle of 30 degrees with the positive x-axis. Find its x and y components.

Solution:

Using the trigonometric formulas for the components of a vector in two dimensions:

v_x = |v| · cosθ

v_y = |v| · sinθ

Given that ( |v| = 5 ) and θ = 30°

v_x = 5 · cos30°, and

v_y = 5 · sin30°

⇒ v_x = 5 \cdot \frac{\sqrt{3}}{2}, and

v_y = 5 × 1/2

Simplifying further:

⇒ v_x = \frac{5\sqrt{3}}{2} , and

v_y = 5/2

So, the x and y components of the vector (v) are \frac{5\sqrt{3}}{2} and 5/2units, respectively.

Example 2: If a = (2, 5) and b = (-1, 3), find the sum a + b in the component form.

Solution:

If a= (2, 5) and b = (-1, 3), find the sum a+b in component form.

The sum of two vectors is obtained by adding their corresponding components. So, for a+b:

a + b = (2 + (-1), 5 + 3)

⇒ a + b= (1, 8)

Therefore, a+b in component form is (1, 8).

Practice Questions on Components of a Vector

Question 1: Consider a vector (v) with a magnitude of 5 units and an angle of 45° with the positive x-axis. Find its rectangular components, vx and vy.

Question 2: Given a three-dimensional vector u = (3, -2, 6), determine its x, y, and z components.

Question 3: If a = (2, 4) and b= (-1, 3), find the sum (a+b) using the algebraic method.

Question 4: Convert the polar vector p with magnitude 8 units and an angle of 60° to its rectangular components.

Answer Key

1. v_x = 3.54 units, v_y = 3.54 units
2. x = 3, y = −2, z = 6
3. a + b = (1, 7)
4. p_x = 4, p_y = 6.93 units