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### Course: Precalculus (Eureka Math/EngageNY)>Unit 2

Lesson 3: Topic D: Lessons 17-18: Vectors in the coordinate plane

# Scalar multiplication: component form

Sal explains what happens graphically and to the components of a vector when we multiply it by a scalar.

## Want to join the conversation?

• Aren't numbers actually vectors because they have their absolute value as magnitude and a direction behind or ahead of zero?
• No, numbers by themselves are not vectors. Being behind or ahead of zero doesn't imply direction, unless you've explicitly started at zero. For example, 5 is 5 greater than zero. But it can also be 5 fewer than 10. So unless there's an origin and an end-point, you don't have a vector.

Besides, numbers can be just abstractions of quantities. If I have five apples, and the five is a vector, what direction does the five in my "five apples" show? Considering numbers to be automatically vectors leads to meaningless questions like that.
• If the scalar is a fraction, would the magnitude change?
• The magnitude would change. For example a vector with the magnitude (4,2) multiplied by the scalar (1/2) would have the magnitude (2,1).
• Multiply positive scalar = change magnitude, keep direction
Multiply negative scalar = change both magnitude and direction

Does this rule always apply when multiplying scalars to vectors, or is it only because he was particularly using -2 and 3?
• It's exactly analogous to multiplying negative and positive numbers. So in this case, yes.
• How is a scalar quantity able to change the direction when it itself only contains magnitude as shown at 4.47
• A scalar can only reverse the direction of a vector (i.e. by multiplying the vector by a negative scalar) but it cannot change the angle other than 180 degrees.
• How is the last vector the same if the direction has changed?
• It isn't, the product of the scalar ,-2 , and vector ,w , create a new vector. If the scalar is negative the direction of the new vector will be the opposite of the original vector though.
• can you use any scalar wile multiplying with a vector?
• Yes, any scalar can be used when multiplying with a vector.
• Since the instructor introduced the idea that we can change the magnitude by multiplying a vector with a positive scalar and we can change both the magnitude and the direction by multiplying a vector with a negative scalar, is it possible to just change the direction of a vector? If so, how (excluding multiplying by -1)?
• Yep, you would need to multiply by a matrix. For instance ultiplying the 2x2 matrix:

0 1
1 0

by a vector swaps the vector's x and y values. So multiplying that matrix by a vector like <2, 3> would make the vector <3,2> You do have to be a little careful witht he matrix to ensure the seze doesn't change.

If you did not learn about matrix multiplication yet you will eventually, or I could try and explain it.
(1 vote)
• How did Sal get 3 for the scalar?
• It was just an example. He could've chosen 5 or 12 or -7, it was just to show how scalar multiplication works
• Can a scalar change the direction of the vector??,when it is multiplied to a vector. And as the vector is both the direction and the vector. Can we say that the scalar when multiplied to a vector can change the vector. As in the `-2w` example which SAL gave at the end of the video