Vectors

Cheat Sheet · Grade 11 Math
Emil Oliversen
WHAT IS A VECTOR?
Magnitude + Direction · Drawn as an arrow
ScalarVector
SpeedVelocity
DistanceDisplacement
MassForce
TemperatureAcceleration
NOTATION
Vector
AB⃗ or bold v
Magnitude
‖v‖ or |v|
Components
[x, y]
COMPASS DIRECTIONS
East
0° (starting point)
North
90°
West
180°
South
270°
40° S of E = 360°−40° = 320°
W 50° S = 180°+50° = 230°
COMPONENTS & MAGNITUDE
x-component
x = ‖v‖ · cos(β)
y-component
y = ‖v‖ · sin(β)
Magnitude
‖v‖ = √(x² + y²)
Angle
β = arctan(y/x)
OPERATIONS
Add
[x₁+x₂, y₁+y₂]
Subtract
[x₁−x₂, y₁−y₂]
Scalar mult
k·[x,y] = [kx, ky]
SPECIAL VECTORS
Opposite
-v = [−x, −y]
Equivalent
Same magnitude & direction
Orthogonal
Dot product = 0
Collinear
v = k·u (scalar multiple)
SCALAR (DOT) PRODUCT
Result is a NUMBER, not a vector
Components
u·v = x₁x₂ + y₁y₂
Magnitudes
u·v = ‖u‖‖v‖cos(θ)
Find angle
cos(θ) = (u·v)/(‖u‖‖v‖)
DOT PRODUCT CASES
u·v = 0
Orthogonal (θ = 90°)
u·v > 0
Acute angle (θ < 90°)
u·v < 0
Obtuse angle (θ > 90°)
Example: u=[6,8] v=[5,12]
u·v = 30+96 = 126
‖u‖=10, ‖v‖=13
θ = arccos(126/130) ≈ 14.5°
APPLICATION STEPS
  • Sketch with labels
  • Break into components
  • Add/subtract components
  • Convert back to magnitude + angle
COMMON MISTAKES
  • Add COMPONENTS not magnitudes
  • Dot product = scalar, not vector
  • Check radians vs degrees in cos
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