Components of a Vector 

The components of a vector are the projections of a vector along the coordinate axes, typically expressed in terms of horizontal and vertical parts in 2D or x, y, and z axes in 3D. Breaking a vector into its components simplifies vector analysis in physics and mathematics. This approach helps in solving complex problems involving direction, motion, and force. Understanding vector components is essential for applications in geometry, engineering, and navigation systems.

1.0Components of a Vector Definition

A vector is a quantity that has both magnitude and direction. The components of a vector are the projections of that vector along the coordinate axes (usually the x-axis and y-axis in 2D, and x, y, z in 3D).

In simple terms, breaking a vector into its parts along each axis gives its vector components.

2.0What Are the Components of a Vector ($\vec{V}$) ?

If you have a vector $\vec{A}$ making an angle θ with the horizontal axis and having a magnitude $|\vec{V}|$, then the components of the vector are:

$\overrightarrow{V_x}=|\vec{V}|\cos \theta ,\overrightarrow{V_y}=|\vec{V}|\sin \theta$

Thus:

$\vec{V}=\overrightarrow{V_x}\hat{i}+\overrightarrow{V_y}\hat{j}$

These are the horizontal and vertical components of $\vec{V}$.

3.0Components of a Vector Formula

In 2D:

If $\vec{V}$ has a magnitude V and makes an angle θ with the x-axis:

$\overrightarrow{V_x}=V\cos (\theta ),\overrightarrow{V_y}=V\sin (\theta )$

In 3D:

If $\overrightarrow{V_x}=<a,b,c>$, then its components are:

• ${\vec{V}}_x=a\hat{i}$
• ${\vec{V}}_y=b\hat{j}$
• ${\vec{V}}_z=c\hat{k}$

4.0What is a 4-Component Vector?

A 4-component vector includes four values, typically represented in higher mathematics and physics (e.g., spacetime vectors in relativity). It takes the form: $\vec{V}=(v_1,v_2,v_3,v_4)$

Each value corresponds to a separate axis or dimension.

5.0Components of a Vector Applications

Vector components are used in:

• Physics: Resolving forces, motion, and velocity into horizontal and vertical directions.
• Engineering: Breaking loads into simpler parts.
• Computer Graphics: Representing direction and motion.
• Robotics: Describing movement in multi-dimensional space.
• Navigation: Determining direction and distance using vectors.

6.0How to Find Components of a Vector: Step-by-Step

1. Find the magnitude of the vector if not given.
2. Measure or use the angle it makes with the reference axis.
3. Apply the formulas: $\overrightarrow{V_x}=|\vec{V}|\cos (\theta ),\overrightarrow{V_y}=|\vec{V}|\sin (\theta )$ 
4. Combine using unit vectors: $\vec{V}=\overrightarrow{V_x}\hat{i}+\overrightarrow{V_y}\hat{j}$

7.0Components of a Vector Examples

Example 1: A vector $\vec{A}$ has magnitude 10 units and angle $30^o$. Find its components.

Solution:

$\overrightarrow{A_x}=10\cos (30^{\circ })=10\cdot \frac{\sqrt{3}}{2}=5\sqrt{3}$

$\overrightarrow{A_y}=10\sin (30^{\circ })=10\cdot \frac{1}{2}=5$

So, $\vec{A}=5\sqrt{3}\hat{i}+5\hat{j}$

Example 2: Given $\vec{B}=<2,-3,4>$. Find its components.

Solution:

• Along x-axis: 2
• Along y-axis: -3
• Along z-axis: 4

Thus, $\vec{B}=2\hat{i}-3\hat{j}+4\hat{k}$

Example 3: A vector $\vec{A}$ has a magnitude of 10 units and makes an angle of $60^o$ with the x-axis. Find its components.

Solution: 

Using the formula:

$A_x=A\cos \theta =10\cos 60^{\circ }=10\cdot \frac{1}{2}=5$

 $A_y=A\sin \theta =10\sin 60^{\circ }=10\cdot \frac{\sqrt{3}}{2}=5\sqrt{3}$

Answer: $\vec{A}=5\hat{i}+5\sqrt{3}\hat{j}$

Example 4: Components of a 3D Vector

Problem:
Find the x, y, and z components of $\vec{B}=<4,-2,6>$

Solution:

The components are:

• $B_x=4$
• $B_y=-2$
• $B_z=6$

Answer: $\vec{B}=4\hat{i}-2\hat{j}+6\hat{k}$

Example 5: A vector has components $V_x=6$ and $V_y=8$. Find the magnitude and direction.

Solution:

Magnitude:

$|\vec{V}|=\sqrt{V_x^2+V_y^2}=\sqrt{6^2+8^2}=\sqrt{100}=10$

Direction:

$\theta ={\tan }^{-1}\left(\frac{V_y}{V_x}\right)={\tan }^{-1}\left(\frac{8}{6}\right)={\tan }^{-1}\left(\frac{4}{3}\right)\approx 53.13^{\circ }$

Answer: Magnitude = 10 units, Direction $\approx 53.13^{\circ }$

Example 6: Given $\vec{F}=3\hat{i}-4\hat{j}$, find its magnitude and angle with the x-axis.

Solution:

Magnitude: $|\vec{F}|=\sqrt{3^2+(-4{)}^2}=\sqrt{9+16}=\sqrt{25}=5$

Angle: $\theta ={\tan }^{-1}\left(\frac{-4}{3}\right)\approx -53.13^{\circ }$

Answer: Magnitude = 5, Angle ≈ $-53.13^{\circ }$

Example 7: A force of 50 N is applied at an angle of $30^o$ above the horizon. Find its horizontal and vertical components.

Solution:

$F_x=50\cos 30^{\circ }=50\cdot \frac{\sqrt{3}}{2}\approx 43.3\text{N}$

$F_y=50\sin 30^{\circ }=50\cdot \frac{1}{2}=25\text{N}$

Answer: $F_x\approx 43.3\text{N},F_y=25\text{N}$

Example 8: Find the components of the vector $\vec{V}$ whose magnitude is 10 and which makes angles of $60^o,45^o,60^o$ with the x-, y-, and z-axes, respectively.

Solution:
Components:

$A_x=10cos60^o=10.\frac{1}{2}=5$

$A_y=10cos45^o=10\frac{1}{\sqrt{2}}=\frac{10}{\sqrt{2}}$

$A_z=10cos60^o=5$

Answer: $A_x=5,A_y=\frac{10}{\sqrt{2}},A_z=5$

 Example 9: Express the vector with initial point at (1, 2, 3) and terminal point at (4, 6, 8) in component form.

Solution:
Components:

$\vec{A}=(4-1)\hat{i}+(6-2)\hat{j}+(8-3)\hat{k}$

$\vec{A}=3\hat{i}+4\hat{j}+5\hat{k}$

Answer: $\vec{A}=3\hat{i}+4\hat{j}+5\hat{k}$

Example 10: Find the projection of $\vec{A}=2\hat{i}+3\hat{j}$ onto $\vec{B}=\hat{i}+\hat{j}$.

Solution:
Formula:

Projection of $\vec{A}and\vec{B}$= $\frac{\vec{A}.\vec{B}}{|\vec{B}|}$

Dot product: $\vec{A}\cdot \vec{B}=2\cdot 1+3\cdot 1=5$

Magnitude of $\vec{B}$: $|\vec{B}|=\sqrt{1^2+1^2}=\sqrt{2}$

Projection:$\frac{5}{\sqrt{2}}=\frac{5\sqrt{2}}{2}$

Answer: $\frac{5\sqrt{2}}{2}$

Example 11: A vector has components 4, 4, 2. Find its direction cosines.

Solution:
Magnitude:$|\vec{A}|=\sqrt{4^2+4^2+2^2}=\sqrt{16+16+4}=\sqrt{36}=6$

Direction cosines: $l=\frac{4}{6}=\frac{2}{3},m=\frac{4}{6}=\frac{2}{3},n=\frac{2}{6}=\frac{1}{3}$

Answer: $l=\frac{2}{3},m=\frac{2}{3},n=\frac{1}{3}$

Example 12: Decompose vector $\vec{A}=5\hat{i}+12\hat{i}$ along the directions of $\vec{B}=\hat{i}$ and $\vec{C}=\hat{j}$.

Solution:
Component along $\vec{B}$: 5 $(\text{coefficient of})\hat{i}$

Component along $\vec{C}$: 12 $\text{(Coefficient of)}\hat{j}$

Answer: 5 along $\vec{B}$, 12 along $\vec{C}$

Example 13: Vectors $\vec{A}$ and $\vec{B}$ have magnitudes 3 and 4, respectively. Find the minimum value of $|\vec{A}-\vec{B}|$.

Solution:
Minimum value when they are in same direction:

$|\vec{A}-\vec{B}{|}_{min}=|3-4|=1$

Answer: 1

Example 14: A force of 10 N acts at $30^o$ to an inclined plane. Find its component along the plane.

Solution:
Component: $10\cos 30^{\circ }=10\cdot \frac{\sqrt{3}}{2}=5\sqrt{3}$

Answer: $5\sqrt{3}$

Example 15: Find the angle between $\vec{A}=\hat{i}+2\hat{j}+2\hat{k}$ and $\vec{B}=2\hat{i}+\hat{j}+2\hat{k}$.

Solution:
Dot product: $1\cdot 2+2\cdot 1+2\cdot 2=2+2+4=8$

Magnitudes:

$|\vec{A}|=\sqrt{1^2+2^2+2^2}=\sqrt{1+4+4}=\sqrt{9}=3$

$|\vec{B}|=\sqrt{2^2+1^2+2^2}=\sqrt{4+1+4}=\sqrt{9}=3$

Angle:

$\cos \theta =\frac{8}{3\cdot 3}=\frac{8}{9}\Rightarrow \theta ={\cos }^{-1}\left(\frac{8}{9}\right)$

Answer: $\theta ={\cos }^{-1}\left(\frac{8}{9}\right)$

Example 16: Find the unit vector in the direction of $\vec{A}=2\hat{i}-\hat{j}+2\hat{k}$.

Solution:
Magnitude:

$|\vec{A}|=\sqrt{2^2+(-1{)}^2+2^2}=\sqrt{4+1+4}=3$

Unit vector: 

$\frac{1}{3}(2\hat{i}-\hat{j}+2\hat{k})=\frac{2}{3}\hat{i}-\frac{1}{3}\hat{j}+\frac{2}{3}\hat{k}$

Answer: $\frac{2}{3}\hat{i}-\frac{1}{3}\hat{j}+\frac{2}{3}\hat{k}$

Example 17: Resolve $\vec{A}=5\hat{i}+5\hat{j}$ into components along $45^{\circ }$ and perpendicular to it.

Solution:
Magnitude:$|\vec{A}|=\sqrt{5^2+5^2}=5\sqrt{2}$

Component along $45^o$: $5\sqrt{2}\cos 45^{\circ }=5\sqrt{2}\cdot \frac{1}{\sqrt{2}}=5$

Perpendicular component: $5\sqrt{2}\sin 45^{\circ }=5\sqrt{2}\cdot \frac{1}{\sqrt{2}}=5$

Answer: Both components are 5

8.0Practice Questions on Vector Components

1. A vector has magnitude 12 and direction $60^o$. Find its components.
2. Find the components of $\vec{V}=<-5,0,12>$.
3. A force of 50 N is applied at an angle of $45^o$. Find its horizontal and vertical components.
4. If $\overrightarrow{V_x}=6,\overrightarrow{V_y}=8$, find the magnitude of $\vec{V}$.
5. Express $\vec{A}=(3,4)$ in terms of unit vectors $\hat{i},\hat{j}$.