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Vectors

Vectors and Transformations

The length of a vector

  • The magnitude is the length of the vector, written with modulus bars as |a| or |AB|
  • It is always positive, because a length cannot be negative, and the direction plays no part
  • The two components form the short sides of a right-angled triangle with the vector as the hypotenuse, so Pythagoras gives the length
A vector drawn from P to Q on axes, with its horizontal component x and vertical component y drawn in green as the two short sides of a right-angled triangle, so the vector itself is the hypotenuse and its magnitude comes from Pythagoras
Source: Magnitude of a vector by Save My Exams
  • For a column vector with components x and y, the magnitude is √(x² + y²)
    • The magnitude of (8, −6) is √(64 + 36) = √100 = 10
    • The magnitude of (5, 12) is √(25 + 144) = √169 = 13
  • Reading the components off a diagram works the same way: the vector AB below runs 10 across and 4 down, so |AB| = √(10² + 4²) = √116 = 10.8 to 3 significant figures
The vector AB drawn as the hypotenuse of a right-angled triangle whose horizontal side is 10 and whose vertical side is −4, so its magnitude is the square root of 10 squared plus 4 squared
Source: Magnitude of a vector by Save My Exams
  • Leave the answer as a surd where the question asks for an exact value