Lesson 3: Multiplying vectors with scalar quantities

• a new vector can be created by repeatedly adding a vector to itself to create a vector with a larger magnitude
• scalar multiplication: when a vector is multiplied by a scalar (number of times the vector is repeated)
• the product of a vector U and a scalar k is a vector that is k times as long as U
• if A is a vector but k is a scalar, then the product of those 2 is a vector
• vectors can only be parallel if one is a scalar multiple of the other
• the product between a scalar and a vector has to be parallel with the original vector
• if the scalar is positive, the product vector points in the same direction as the original vector
• if the scalar multiple is negative, the product vector points in the opposite direction as the original vector
• when one vector is a scalar multiple of another, these vectors are collinear
• Collinear: when 2 vectors share a line, when put together they form a single line
• when 2 vectors that are non-collinear are added or subtracted from each other, the resultant vector is a linear combination of the 2 vectors
• Commutative property: U + V = V + U (order of adding the vectors doesn't matter)
• Associative property: (U+V)+W = U+(V+W) (no matter the placement of brackets, the result is always the same)
• Additive identity: U+0 = U (adding a vector by a zero vector, so it stays the same)
• Additive inverse: U+(-U) = 0 (a vector adding the inverse of itself equals a zero vector)
• Scalar over vector sum: m(U+V) = mU + mV
• Vector over scalar sum: (m+n)U = mU + nU
• associative property for multiplication: (mn)U = m(nU) = n(mU) (the grouping doesn't matter because the answer will still be the same)
• multiplicative identity: 1U = U (multiplying a vector by one doesn't change it)
• Properties of the Zero vector:

  1. U-U = 0 (drawn as a dot)
  2. U+0 = U (adding 0 to a vector keeps the same value)
  3. m0 = 0 vector
  4. 0U = 0 vector (if the multiple is 0, the resultant vector is the 0 vector)