2.7 Vector Components

Definition

A vector can be resolved into two perpendicular component vectors acting along the coordinate axes. These vectors are called the components of the original vector.

In a two-dimensional Cartesian coordinate system, every vector has:

  • a horizontal component (Ax)
  • a vertical component (Ay)

Together, these component vectors are equivalent to the original vector.

Resolving a Vector into Components

Figure 2.7.1 – A vector resolved into its horizontal and vertical components.

From the right triangle formed by the vector and its components,

Ax=Acos⁡θA_x = A\cos\theta

Ay=Asin⁡θA_y = A\sin\theta

where

  • A is the magnitude of the vector,
  • Aₓ is the horizontal component,
  • Aᵧ is the vertical component,
  • θ is measured from the positive x-axis.

The horizontal component is adjacent to the angle, while the vertical component is opposite the angle.

Each component is itself a vector because it has both magnitude and direction.

Component Representation

Once the vector components are known, the vector can be written as

A→=(Ax,Ay)\vec{A} = (A_x, A_y)

For example,

A→=(3,4)\vec{A} = (3, 4)

means the vector extends

  • 3 units along the positive x-axis,
  • 4 units along the positive y-axis.

The signs of the components determine the direction of the vector. For example,

VectorDirection
(3, 4)right and up
(-3, 4)left and up
(-3, -4)left and down
(3, -4)right and down
Table 2.7.1 – Direction of a vector based on the signs of its components.

Example

A force of 10 N acts at 30° above the horizontal. Using the component equations,

Fx=10cos⁡30∘=8.66NF_x = 10\cos 30^\circ = 8.66\,\mathrm{N}

Fy=10sin⁡30∘=5.00NF_y = 10\sin 30^\circ = 5.00\,\mathrm{N}

The force can therefore be written as

F→=(8.66,5.00)N\vec{F} = (8.66,\,5.00)\,\mathrm{N}

Finding the Magnitude and Direction from Vector Components

If the vector components are known, its magnitude and direction can be determined.

Its magnitude is

A=Ax2+Ay2A = \sqrt{A_x^2 + A_y^2}

Its direction is

θ=tan−1⁡(AyAx)\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)

These equations are simply the reverse of resolving a vector into components.

Example: Finding the Magnitude and Direction of a Vector

Figure 2.7.1 shows a vector resolved into its horizontal and vertical components. Suppose that the horizontal component is 3 units and the vertical component is 4 units.

Determine:

  • the magnitude of the vector,
  • the direction of the vector measured from the positive x-axis.

Solution:

The magnitude of the vector is found using the Pythagorean theorem.

A=Ax2+Ay2A = \sqrt{A_x^2 + A_y^2}

A=32+42A = \sqrt{3^2 + 4^2}

A=9+16A = \sqrt{9 + 16}

A=25A = \sqrt{25}

A=5unitsA = 5\,units

Next, determine the direction using the tangent ratio.

θ=tan−1⁡(AyAx)\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)

θ=tan−1⁡(43)\theta = \tan^{-1}\left(\frac{4}{3}\right)

θ=53.1∘\theta = 53.1^\circ

Answer

Magnitude: 5 units

Direction: 53.1° above the positive x-axis.

Key Points

  • Every two-dimensional vector can be resolved into horizontal and vertical components.
  • The components are perpendicular to each other.
  • Components can be calculated using sine and cosine, while the original vector can be reconstructed using the Pythagorean theorem and the inverse tangent function.
  • A vector can be represented by its component form (Aₓ, Aᵧ).
  • The original vector can always be reconstructed from its components.

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2.8 Unit Vectors →