Vector Coordinates
A vector is a quantity that has both magnitude and direction. Unlike scalars, which only have size, vectors also tell us the direction in which the quantity acts.
- Representation: Vectors can be drawn as arrows (geometric form) or written in coordinate form (algebraic).
- Components: In 2D, vectors are expressed using the x and y axes, while in 3D they also include the z axis.
- Geometric Meaning: Vectors describe displacement, velocity, acceleration, and forces.
- Applications: Physics (motion and mechanics), navigation (aircraft, ships), and engineering graphics.
Vector Addition
Vector addition combines two or more vectors to form a resultant.
- Graphical Methods: Triangle law and parallelogram law of vector addition.
- Properties: Vector addition is commutative (order doesn’t matter) and associative (grouping doesn’t matter).
- Practical Examples: Combining forces acting on an object or finding the net displacement of a traveler.
Vector Subtraction
Vector subtraction gives the difference between two vectors.
- Concept: Subtracting one vector means adding its negative.
- Geometrical Interpretation: The result points from the tip of one vector to the tip of another.
- Applications: Relative velocity between two moving objects, displacement comparison.
Scaling Vectors
Scaling involves multiplying a vector by a scalar (a real number).
- Effect: The magnitude changes (stretching or shrinking), while direction remains the same unless multiplied by a negative, which reverses it.
- Real-life Uses: Adjusting the force applied, scaling velocities in simulations, or magnifying displacements in graphics.
Matrices and Determinants Formulas | Properties, Operations & Applications
Scalar Product (Dot Product)
The dot product of two vectors results in a scalar (number).
- Concept: It measures how much one vector aligns with another.
- Applications: Calculating work done by a force, projecting one vector onto another, measuring similarity of directions.
Vector Product (Cross Product)
The cross product of two vectors results in another vector.
- Perpendicular Nature: The new vector is perpendicular to both original vectors.
- Applications: Torque in physics, rotational motion, finding the area of a parallelogram defined by two vectors.
Triple Product
There are two types:
- Scalar Triple Product: Gives a scalar representing the volume of a parallelepiped formed by three vectors.
- Vector Triple Product: Produces a vector by taking the cross product of one vector with the cross of two others.
- Applications: Used in 3D geometry, mechanics, and analyzing space structures.
