Vectors are the building blocks of geometry in motion. They pack a punch with both size and direction, letting us map out forces and movements in space. Think of them as arrows that point the way and tell us how far to go.
Adding and subtracting vectors is like playing connect-the-dots. We can join them tip-to-tail or use math to combine their parts. Multiplying by numbers stretches or shrinks vectors, giving us new tools to solve real-world problems.
Vector Fundamentals
Properties of geometric vectors
Top images from around the web for Properties of geometric vectors Vectors | Boundless Physics View original
Is this image relevant?
Vectors in the Plane · Calculus View original
Is this image relevant?
Scalars and Vectors – University Physics Volume 1 View original
Is this image relevant?
Vectors | Boundless Physics View original
Is this image relevant?
Vectors in the Plane · Calculus View original
Is this image relevant?
1 of 3
Top images from around the web for Properties of geometric vectors Vectors | Boundless Physics View original
Is this image relevant?
Vectors in the Plane · Calculus View original
Is this image relevant?
Scalars and Vectors – University Physics Volume 1 View original
Is this image relevant?
Vectors | Boundless Physics View original
Is this image relevant?
Vectors in the Plane · Calculus View original
Is this image relevant?
1 of 3
Vectors possess both magnitude and direction
Magnitude signifies the length of the vector (distance between initial and terminal points)
Direction specifies the orientation of the vector in space (angle relative to a reference axis)
Vectors are graphically depicted as arrows in geometric contexts
Arrow tail marks the initial point (starting position)
Arrow head indicates the terminal point (ending position)
Vectors are considered equivalent when they have identical magnitudes and directions, irrespective of their spatial locations
Translating a vector parallel to itself does not alter its properties
Parallel vectors share the same direction but may differ in magnitude
Scaling a vector changes its length while maintaining its direction
Zero vector has a magnitude of zero and lacks a specific direction
Represented by a point rather than an arrow
Vector Operations
Vector addition and subtraction
Graphical vector addition
Place the tail of the second vector at the head of the first vector to add them graphically
Resultant vector extends from the tail of the first vector to the head of the second vector (triangle law of vector addition)
Algebraic vector addition
Express vectors as ordered pairs ( a , b ) (a, b) ( a , b ) in a 2D coordinate system
Add corresponding components to perform algebraic vector addition: ( a 1 , b 1 ) + ( a 2 , b 2 ) = ( a 1 + a 2 , b 1 + b 2 ) (a_1, b_1) + (a_2, b_2) = (a_1 + a_2, b_1 + b_2) ( a 1 , b 1 ) + ( a 2 , b 2 ) = ( a 1 + a 2 , b 1 + b 2 )
Graphical vector subtraction
Reverse the direction of the vector being subtracted and add it to the other vector for graphical subtraction
Resultant vector extends from the tail of the first vector to the head of the reversed second vector
Algebraic vector subtraction
Subtract corresponding components to perform algebraic vector subtraction: ( a 1 , b 1 ) − ( a 2 , b 2 ) = ( a 1 − a 2 , b 1 − b 2 ) (a_1, b_1) - (a_2, b_2) = (a_1 - a_2, b_1 - b_2) ( a 1 , b 1 ) − ( a 2 , b 2 ) = ( a 1 − a 2 , b 1 − b 2 )
Scalar multiplication of vectors
Scalar multiplication involves multiplying a vector by a real number (scalar) to produce a new vector
Multiplying vector v ⃗ \vec{v} v by scalar c c c yields a new vector c v ⃗ c\vec{v} c v
Scalar multiplication alters the magnitude of the vector
∣ c ∣ > 1 |c| > 1 ∣ c ∣ > 1 results in a longer vector than the original
0 < ∣ c ∣ < 1 0 < |c| < 1 0 < ∣ c ∣ < 1 results in a shorter vector than the original
Scalar multiplication can modify the direction of the vector
c > 0 c > 0 c > 0 preserves the original vector's direction
c < 0 c < 0 c < 0 reverses the original vector's direction
Scalar multiplication by zero yields the zero vector
Magnitude and direction in plane
The magnitude of a vector v ⃗ = ( a , b ) \vec{v} = (a, b) v = ( a , b ) , denoted as ∣ v ⃗ ∣ |\vec{v}| ∣ v ∣ , is calculated using the Pythagorean theorem: ∣ v ⃗ ∣ = a 2 + b 2 |\vec{v}| = \sqrt{a^2 + b^2} ∣ v ∣ = a 2 + b 2
Magnitude represents the length of the vector
The direction of a vector is described by the angle it forms with the positive x-axis
Angle θ \theta θ is calculated using the arctangent function: θ = tan − 1 ( b a ) \theta = \tan^{-1}(\frac{b}{a}) θ = tan − 1 ( a b ) , where a a a and b b b are the vector's components
Unit vectors have a magnitude of 1 and solely represent direction
To find the unit vector in the same direction as v ⃗ \vec{v} v , divide the vector by its magnitude: v ^ = v ⃗ ∣ v ⃗ ∣ \hat{v} = \frac{\vec{v}}{|\vec{v}|} v ^ = ∣ v ∣ v
Unit vectors are useful for describing directions without considering magnitude