Vector Calculator
Scalars and Vectors
In physics and mathematics, quantities are often classified as either scalars or vectors.
- A scalar is a quantity that is fully described by its magnitude (or numerical value ) alone. Examples: distance (5 km), speed (60 km/h), mass (10 kg), temperature (25°C).
- A vector is a quantity that has both magnitude and direction. Examples: displacement (5 km North), velocity (60 km/h East), force (10 N downwards), acceleration (9.8 m/s² downwards).
Vector Representation
Vectors are typically represented in several ways:
- Graphically: As an arrow. The length of the arrow represents the vector's magnitude, and the direction of the arrow (indicated by the arrowhead) represents its direction.
- Symbolically:
- Boldface letter: $\mathbf{v}$ or $\mathbf{F}$
- Letter with an arrow above: $\vec{v}$ or $\vec{F}$
- Magnitude of a vector $\mathbf{v}$ is written as $|\mathbf{v}|$ or $v$.
- Component Form (2D): A vector $\mathbf{v}$ can be expressed by its horizontal ($v_x$) and vertical ($v_y$) components as $(v_x, v_y)$ or $v_x\mathbf{i} + v_y\mathbf{j}$.
- Component Form (3D): A vector $\mathbf{v}$ with components $V_x, V_y,$ and $V_z$ is written as $\mathbf{V} = V_x\mathbf{i} + V_y\mathbf{j} + V_z\mathbf{k}$. For example, if $\mathbf{B} = (2, -1, 5)$, then $\mathbf{B} = 2\mathbf{i} - \mathbf{j} + 5\mathbf{k}$.
Graphical Vector Addition (2D)
Visualize $\mathbf{A} + \mathbf{B}$
Enter two vectors $\mathbf{A} = (A_x, A_y)$ and $\mathbf{B} = (B_x, B_y)$.
Vector A Components
Vector B Components
Resolving a Vector into Components
Find $V_x$ and $V_y$ from Magnitude $|V|$ and Angle $\theta$
Vector Addition by Calculation
Sum of Two Vectors $\mathbf{A} + \mathbf{B}$
Enter vectors as components: $\mathbf{A} = (A_x, A_y)$ and $\mathbf{B} = (B_x, B_y)$.
Vector A Components
Vector B Components
Vector Subtraction
Vector subtraction $\mathbf{A} - \mathbf{B}$ is defined as the addition of $\mathbf{A}$ and the negative of $\mathbf{B}$: $$ \mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B}) $$ If $\mathbf{B} = (B_x, B_y)$, then $-\mathbf{B} = (-B_x, -B_y)$. The vector $-\mathbf{B}$ has the same magnitude as $\mathbf{B}$ but points in the opposite direction.
Difference of Two Vectors $\mathbf{A} - \mathbf{B}$
Enter vectors as components: $\mathbf{A} = (A_x, A_y)$ and $\mathbf{B} = (B_x, B_y)$.
Vector A Components
Vector B Components
Relative Velocity
Relative velocity is the velocity of an object as observed from a particular frame of reference, which itself may be moving. If $\mathbf{v}_A$ is the velocity of object A and $\mathbf{v}_B$ is the velocity of object B (both relative to a common stationary frame, like the ground), then:
- The velocity of A relative to B is $\mathbf{v}_{AB} = \mathbf{v}_A - \mathbf{v}_B$.
- The velocity of B relative to A is $\mathbf{v}_{BA} = \mathbf{v}_B - \mathbf{v}_A = - \mathbf{v}_{AB}$.
Calculate Velocity of A relative to B ($\mathbf{v}_{AB}$)
Enter velocities as components: $\mathbf{v}_A = (v_{Ax}, v_{Ay})$ and $\mathbf{v}_B = (v_{Bx}, v_{By})$.
Velocity of A ($\mathbf{v}_A$) Components
Velocity of B ($\mathbf{v}_B$) Components
$\mathbf{i}, \mathbf{j}, \mathbf{k}$ Notation
Vectors can be expressed in terms of unit vectors $\mathbf{i}, \mathbf{j},$ and $\mathbf{k}$. These are vectors of magnitude 1 pointing along the positive x, y, and z axes, respectively.
- In 2D: A vector $\mathbf{V}$ with components $V_x$ and $V_y$ is written as $\mathbf{V} = V_x\mathbf{i} + V_y\mathbf{j}$. For example, if $\mathbf{A} = (3, 4)$, then $\mathbf{A} = 3\mathbf{i} + 4\mathbf{j}$.
- In 3D: A vector $\mathbf{V}$ with components $V_x, V_y,$ and $V_z$ is written as $\mathbf{V} = V_x\mathbf{i} + V_y\mathbf{j} + V_z\mathbf{k}$. For example, if $\mathbf{B} = (2, -1, 5)$, then $\mathbf{B} = 2\mathbf{i} - \mathbf{j} + 5\mathbf{k}$.
Vector operations using this notation:
- Addition: $(A_x\mathbf{i} + A_y\mathbf{j}) + (B_x\mathbf{i} + B_y\mathbf{j}) = (A_x+B_x)\mathbf{i} + (A_y+B_y)\mathbf{j}$
- Scalar Multiplication: $k(A_x\mathbf{i} + A_y\mathbf{j}) = (kA_x)\mathbf{i} + (kA_y)\mathbf{j}$