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The operation of multiplying a vector by a scalar is called scalar multiplication. This consists of multiplying each component of the two- or three-dimensional vector by the scalar.
k* ⟨ a,b ⟩ &= ⟨ k * a, k * b ⟩ & or k* ⟨ a,b,c ⟩ &= ⟨ k * a, k * b, k* c ⟩
Depending on the value of the scalar, the vector can be reduced or enlarged. If the absolute value of the scalar is less than 1, the vector is reduced. Conversely, if the absolute value of the scalar is greater than 1, the vector is enlarged. For example, consider the vector v = ⟨ 3,2 ⟩ and see how it is affected when multiplied by a positive scalar.
Notice that 2v has the same direction as v but is twice as long. Similarly, 0.5v has the same direction as v but is half as long. When the scalar is negative, the direction of the new vector is opposite to the direction of the original vector. In the example below, see what happens when u=⟨ -2,3 ⟩ is multiplied by negative scalars.
The vector -2u is twice as long as u but it has the opposite direction. Furthermore, -0.75u also has the opposite direction as u but its length is 0.75 times the length of u.