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Physics Lesson 2.3.3 - Multiplying a vector by a negative scalar
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Welcome to our Physics lesson on Multiplying a vector by a negative scalar, this is the third lesson of our suite of physics lessons covering the topic of Multiplication of a Vector by a Scalar, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
Multiplying a vector by a negative scalar
It is obvious that when we multiply a (positive) vector by a negative scalar, this results in a negative vector. This is because (+) × (-) = (-). Therefore, when multiplying the vector u⃗ by a negative number N, the result will be still v⃗ = N × u⃗ but the direction of v⃗ will be the opposite of u⃗. For example, if N = -2, the vector v⃗ is twice as long as the vector u⃗ but these vectors have opposite direction. Look at the figure below:
From the figure, it is easy to notice that when a vector is obtained by multiplying another vector by a negative scalar, the two vectors in question are antiparallel (i.e. parallel but with opposite direction).
The same is true for the division of a vector by a scalar as well. The two vectors will have again opposite direction. The only difference is that the second (output) vector will be smaller in length than the first (input) one.
Example 3
An object moves linearly by 24 m due West. Then it moves in the opposite direction by one third the magnitude of the first displacement.
What is the second displacement of the object in metres?
Show the two vectors in a figure by making a suitable sketch.
What is the final position of the object if we take as zero the initial position (starting point)?
Solution 3
a. This is a typical example of the division of a vector by a negative scalar where the final vector will have an opposite direction to the original one. Since the first displacement ∆x⃗1 is due West, the second displacement ∆x⃗2 will be due East. We can write:
Or
Thus, the second displacement is 8m due East.
b. The situation is graphically represented through the figure below.
c. From the figure, it is obvious the object's final position (which represents the total displacement Δx⃗tot is calculated by adding the two displacements (see the topic "Addition and Subtraction of Vectors"). Thus, we have
= 24m + (-8m)
= 16m
Therefore, at the end of motion the object will be at 16 m due East to the original position.
You have reach the end of Physics lesson 2.3.3 Multiplying a vector by a negative scalar. There are 4 lessons in this physics tutorial covering Multiplication of a Vector by a Scalar, you can access all the lessons from this tutorial below.
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