Vector Components

Resolving a Vector along the Coordinate Axes

Every vector in the coordinate plane can be decomposed along the two coordinate axes. In $\vec{v}=a\mathbf{i}+b\mathbf{j}$, the signed numbers $a$ and $b$ are the scalar components. The vectors $a\mathbf{i}$ and $b\mathbf{j}$ are the corresponding vector components, or orthogonal projections onto the axes.

The scalar components tell us how far and in which signed direction the vector extends along the $x$- and $y$-axes. Every vector in the plane is a linear combination of the unit vectors $\mathbf{i}$ and $\mathbf{j}$.

If we have a vector $\overrightarrow{AB}$, then:

The components in this form are:

• $a$ is the horizontal scalar component, and $a\mathbf{i}$ is the corresponding vector projection
• $b$ is the vertical scalar component, and $b\mathbf{j}$ is the corresponding vector projection
• $\mathbf{i}$ is the unit vector in the direction of the $x$-axis
• $\mathbf{j}$ is the unit vector in the direction of the $y$-axis

Example of Vector Components

Consider the vector $\overrightarrow{AB}$ in the diagram. This vector can be written as:

This means that $\overrightarrow{AB}$ has scalar components $6$ and $8$, with vector projections $6\mathbf{i}$ to the right and $8\mathbf{j}$ upward.

Vector Magnitude from Its Components

When we know the components of a vector, we can calculate the length or magnitude of the vector using the Pythagorean theorem.

The magnitude of vector $\overrightarrow{AB}$ is denoted by $|\overrightarrow{AB}|$ and calculated using the formula:

In this formula, $a$ and $b$ are the components of $\overrightarrow{AB}$.

Example of Vector Magnitude Calculation

For the vector $\overrightarrow{AB} = 6\mathbf{i} + 8\mathbf{j}$, its magnitude is:

Vectors in Three Dimensional Space

Vectors can be used on a two-dimensional plane and in three-dimensional space.

In three-dimensional space, a vector has three components: the x-component, y-component, and z-component. A vector in three-dimensional space can be expressed as:

The components in this form are:

• $a$ is the scalar component along the $x$-axis, while $a\mathbf{i}$ is its vector projection
• $b$ is the scalar component along the $y$-axis, while $b\mathbf{j}$ is its vector projection
• $c$ is the scalar component along the $z$-axis, while $c\mathbf{k}$ is its vector projection
• $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are unit vectors in the direction of the $x$, $y$, and $z$ axes

The magnitude of a vector in three-dimensional space is calculated using the formula:

Unit Direction Vector

To represent the direction of a nonzero vector, we can use a unit vector. A unit vector has magnitude $1 \text{ unit}$. We obtain it by dividing the nonzero vector by its magnitude.

Denote the unit direction vector of $\overrightarrow{AB}$ by $\mathbf{u}_{AB}$. It is calculated using:

Example of Unit Direction Vector

For the vector $\overrightarrow{AB} = 6\mathbf{i} + 8\mathbf{j}$ with magnitude $10$, its unit direction vector is:

This unit vector indicates the direction of vector $\overrightarrow{AB}$ without regard to its magnitude.

Applications of Vector Components

Resolving a vector along chosen axes supports these calculations:

• Calculating velocity and displacement in physics
• Analyzing forces in mechanics
• Determining the direction and magnitude of resultants in object movement
• Navigation and position determination in coordinate systems

Vector components separate magnitude and direction along chosen axes, so each axis can be calculated independently before the results are combined.