Till here, you must have learned that a vector only has two things, direction, and magnitude. If you change one of those parameters, you will get a whole new vector that is different from its parent vector. To increase or decrease the magnitude of any vector (without changing the direction), we do scalar multiplication. In this lesson, you will understand scalar multiplication and its properties.

Scalar multiplication means to multiply a vector with a scalar number. Actually, the vector is never multiplied by a scalar, it is always multiplied by a vector but then what does scalar multiplication means? It means that you are multiplying the components of a vector by a scalar number. The best part is that it either increases or decreases the magnitude of the vector without changing its direction. It means that the direction of the vector will remain the same but the length of the vector may vary. For example, we have a vector \vec { u }, now if we multiply the vector \vec { u } by the value 3 (which is a scalar number), we will get 3 \vec { u }. It means that the vector magnitude has increased three times the size of the previous vector. Furthermore, you will see an increase in the length of the vector but the direction will remain the same.

Basically, there are two conditions for scalar multiplication. The first one is that the number multiplied to the vector should be a scalar quantity. The second condition is that the direction of the vector will never change if you are performing scalar multiplication. Scalar multiplication will not always result in an increase in magnitude, sometimes it can decrease the value of the original vector for example, if you multiply the vector by \frac { 1 }{ 2 }, the magnitude of the vector will decrease which will ultimately decrease the length of the vector.


A vector \vec { u } = (-2, 5) is to be elevated to (-6, 15). What scalar value will you multiply to convert this vector?

\vec { u } = (-2, 5)

3 \times \vec { u } = 3(-2, 5)

3 \vec { u } = (-6, 15)


Properties of Scalar Multiplication

Property No.1: Associative

It doesn't matter what way you group, it will always result in the same. In simple words, the way in which the vectors are grouped does not change the result.

k . (k' . \vec { u }) = (k . k') . \vec { u }

Property No.2: Right distributivity


k . (\vec { u } + \vec { v }) = k . \vec { u } + k . \vec { v }


Property No.3: Left distributivity


(k + k') . \vec { u } = k . \vec { u } + k'. \vec { u }


Property No.4: Multiplicative identity

This property states that if you multiply any vector with a unit, it will always result in the same vector.

1 . \vec { u } = \vec { u }

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