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Multiplication of a Vector by a Scalar

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2.3Multiplication of a Vector by a Scalar


In these revision notes for Multiplication of a Vector by a Scalar, we cover the following key points:

  • The meaning of "Multiplication of a vector by a scalar"
  • How to express the division of a vector by a scalar in terms of multiplication?
  • What are some examples involving the multiplication of a vector by a scalar in Physics?
  • How to multiply a vector by a scalar in coordinates?

Multiplication of a Vector by a Scalar Revision Notes

If we multiply a vector by a positive scalar, the result will be a new vector whose characteristics (direction, unit, the physical quantity they represent etc.) are all similar to the original vector except length. Thus, if we read in a certain text that CD = 3 × AB we understand three things:

  1. Vectors AB and CD have the same direction (they are parallel or collinear)
  2. Not always, they represent the same type of physical quantity. It only occurs when the scalar has no unit (is dimensionless).
  3. The vector CD is triple in length the vector AB

We can generalize this rule by writing:

If a vector u is parallel to another vector v and the length of u is N times greater than the length of v, then

u= N × v

where N is a number (scalar).

If we have a vector u and we want to obtain a new vector v which has the same direction but is N-times smaller, we can write the equation in two ways:

v = u/N
v = 1/N × u

When multiplying a vector u by a negative number N, the result will be still v = N × u but the direction of v will be the opposite of u. For example, if N = -2, we obtain the equation |v| = -2 × |u| (the vector v is twice as long as the vector u but these vectors have opposite direction.

The same is also true for the division of a vector by a scalar. 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.

To multiply or divide a vector by a scalar in coordinates, we simply multiply or divide each coordinate of the original vector with the given scalar. The result represents the coordinates of the new vector.

Mathematically, we can write:

v = N × u = N × xvyvzv = N × xvN × yvN × zv

The same method can be also used for the multiplication of a vector by a negative scalar of for division of a vector by a positive or negative scalar. This method is much easier than solving the questions graphically.

We hope you found this tutorial useful, if you did. Please take the time to rate this tutorial and/or share on your favourite social network. In our next tutorial, we explore and explain the Dot (Scalar) Product of Two Vectors.

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