Class 11 > Unit # 02: Kinematics > Unit vector, free vector, Null Vector, Position vector

Class 11 Physics • Unit 2 Kinematics

Types of Vectors: Unit, Null, Free, and Position Vectors

Comprehensive lecture notes covering unit vectors in 3D space, null vectors, free vectors, and position vectors with mathematical formulations.

1. Unit Vector

Unit Vector: A vector having a magnitude of exactly unity ($1$) whose sole purpose is to specify the direction of a given vector quantity.

Mathematically, a unit vector in the direction of vector $\vec{A}$ is denoted by $\hat{A}$ (read as "A cap") and is calculated by dividing the vector by its magnitude:

$$\hat{A} = \frac{\vec{A}}{|\vec{A}|}$$

Unit Vectors in 3D Cartesian Coordinate Space ($\hat{i}, \hat{j}, \hat{k}$):

• $\hat{i}$ (i-cap): Unit vector pointing along the positive $X$-axis (Right direction).
• $\hat{j}$ (j-cap): Unit vector pointing along the positive $Y$-axis (Upward direction).
• $\hat{k}$ (k-cap): Unit vector pointing along the positive $Z$-axis (Out of the screen / towards the observer).

Vector Representation: Any general 3D vector can be expressed using unit vectors as: $\vec{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$.

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2. Null Vector (Zero Vector)

Null Vector: A vector having a magnitude of zero ($0$) and an arbitrary or undefined direction.

$$\vec{0} = 0\hat{i} + 0\hat{j} + 0\hat{k}$$

Physical Significance: A null vector naturally appears as the resultant when two equal and opposite vectors are added together ($\vec{A} + (-\vec{A}) = \vec{0}$). It preserves the vector nature of equations where the net vector quantity evaluates to zero.

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3. Free Vector

Free Vector: A vector that can be displaced anywhere in space parallel to itself without changing its magnitude or direction.

Most physical vectors (like velocity or displacement) are free vectors because their spatial position does not affect their physical significance as long as their length and orientation remain identical.

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4. Position Vector

Position Vector: A vector that specifies the position of a point relative to the origin of a reference frame. Unlike free vectors, a position vector is not free and must always start at the origin.

$$\vec{r}_A = \vec{OA}$$

Distance Between Two Points: Using position vectors $\vec{r}_A$ and $\vec{r}_B$ for points $A$ and $B$, the vector distance from $A$ to $B$ is given by vector subtraction:

$$\vec{AB} = \vec{r}_B - \vec{r}_A$$