From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!
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|.