KAHIBARO
Table of Contents
Breaking a Vector into Parts
A vector often points in some direction that is not purely horizontal or purely vertical. Vector components are the separate parts of that vector along chosen coordinate axes. In beginner physics, the most common axes are the $x$ axis and the $y$ axis. Instead of working with one slanted arrow, we replace it with two simpler arrows, one horizontal and one vertical.
This is useful because many physics problems become much easier when each direction is handled separately. Motion, force, and fields are often analyzed one axis at a time.
Components in Two Dimensions
Suppose a vector $\vec{A}$ has magnitude $A$ and makes an angle $\theta$ with the positive $x$ axis. Its horizontal component is called $A_x$, and its vertical component is called $A_y$.
If the angle is measured from the positive $x$ axis, then
$$
A_x = A\cos\theta
$$
$$
A_y = A\sin\theta
$$
These two numbers tell us how much of the vector lies along each axis.
Why Sine and Cosine Appear
The vector and its components form a right triangle. In that triangle,
$$
\cos\theta = \frac{A_x}{A}
\qquad \text{and} \qquad
\sin\theta = \frac{A_y}{A}
$$
So multiplying both sides by $A$ gives the component formulas. This is a direct use of right triangle trigonometry.
Signs of Components
Components are not always positive. A component is positive if it points in the positive axis direction, and negative if it points in the opposite direction.
For example, if a vector points to the left, then its $x$ component is negative. If it points downward, then its $y$ component is negative.
The signs depend on direction, not on the size of the vector.
Important rule:
A component can be negative.
Magnitude is always non negative, but components may be positive, negative, or zero.
Components in Different Quadrants
The angle and direction determine the signs of the components.
| Quadrant | Direction of vector | Sign of $A_x$ | Sign of $A_y$ |
|---|---|---|---|
| I | right and up | $+$ | $+$ |
| II | left and up | $-$ | $+$ |
| III | left and down | $-$ | $-$ |
| IV | right and down | $+$ | $-$ |
For example, a vector in the second quadrant has a negative horizontal component and a positive vertical component.
Reconstructing the Vector from Its Components
If the components are known, we can recover the original vector's magnitude and direction.
The magnitude is found using the Pythagorean theorem:
$$
A = \sqrt{A_x^2 + A_y^2}
$$
The direction angle satisfies
$$
\tan\theta = \frac{A_y}{A_x}
$$
You must be careful with the angle because the tangent ratio alone does not always tell you the correct quadrant. The signs of $A_x$ and $A_y$ must also be checked.
Key formulas for a vector in two dimensions:
$$
A_x = A\cos\theta, \qquad A_y = A\sin\theta
$$
$$
A = \sqrt{A_x^2 + A_y^2}
$$
$$
\tan\theta = \frac{A_y}{A_x}
$$
Example
Let a vector have magnitude $10$ units and direction $\theta = 30^\circ$ above the positive $x$ axis.
Then
$$
A_x = 10\cos 30^\circ = 10\left(\frac{\sqrt{3}}{2}\right) \approx 8.66
$$
$$
A_y = 10\sin 30^\circ = 10\left(\frac{1}{2}\right) = 5
$$
So the vector is equivalent to a horizontal component of about $8.66$ units and a vertical component of $5$ units.
Components in Three Dimensions
In three dimensions, a vector can be split into three perpendicular components:
$$
\vec{A} \to (A_x, A_y, A_z)
$$
These correspond to the parts of the vector along the $x$, $y$, and $z$ axes. The magnitude becomes
$$
A = \sqrt{A_x^2 + A_y^2 + A_z^2}
$$
The main idea is the same as in two dimensions, only with one extra direction.
Physical Meaning
In physics, components let us describe how much of a vector acts in a particular direction. For example, a force pulling at an angle can be separated into a horizontal part and a vertical part. A velocity can also be split into sideways and upward parts. The actual applications are studied in later chapters, but the mathematical idea is always the same, break one vector into independent directional pieces.
A Common Interpretation
A component is not a separate new vector unrelated to the original one. It is one part of the original vector along a chosen axis. Together, the components reproduce the full vector.
Final Idea
Vector components turn a single angled vector into simpler directional pieces. Once a coordinate system is chosen, the vector can be described completely by its components. This is one of the most important tools in physics because many problems become easier when each direction is treated separately.