Scaling Vectors Examples

Multiply the vector <9, 2> by 5.

We multiply each component by 5. The notation goes like this: 5<9, 2> = <5 · 9, 5 · 2> = <45, 10>.

What is the magnitude of the vector 5<2, 3>?

There are two ways to do this. 

Way 1: Do the multiplication first.

5<2,3> = <10, 15>.

The magnitude of this vector is

Way 2: The magnitude of the vector <2, 3> is

If we scale this vector by 5 we also scale its magnitude by 5, therefore

If the vector a<4, 3> has magnitude 10, what is a?

The vector <4, 3> has magnitude

The magnitude of a<4,3> is twice as great as the magnitude of <4, 3>.

We must have multiplied the original vector <4, 3> by 2, so a = 2.

We can also scale vectors to make them shorter, rather than longer.

By what number must we multiply the vector <3, 5> to get a vector of length 2?

The vector <3, 5> has magnitude

If we want to get a vector with length 2, we have to multiply the vector <3, 5> by .

The resulting vector 

will have magnitude 

What happens to the magnitude and direction of the vector <2, 2> when it is multiplied by -1 ?

Do the multiplication and find the new vector: (-1)<2, 2> = <-2, -2>.

The magnitude of the original vector was

The magnitude of the new vector is

The magnitude of the vector didn't change.

The original vector, <2, 2>, had a direction of . The new vector <-2, -2> has a direction of  .

The new vector is traveling in the opposite direction from the old vector.

When we multiply a vector by a negative number a to get a new vector, the negative sign causes the new vector to point in the opposite direction of the original vector. The new vector is scaled by a factor of |a|.