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In addition to the revision notes for Multiplication of a Vector by a Scalar on this page, you can also access the following Vectors and Scalars learning resources for Multiplication of a Vector by a Scalar
Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
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2.3 | Multiplication of a Vector by a Scalar |
In these revision notes for Multiplication of a Vector by a Scalar, we cover the following key points:
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 CD⃗ = 3 × AB⃗ we understand three things:
We can generalize this rule by writing:
If a vector u⃗ is parallel to another vector v⃗ and the length of u⃗ is N times greater than the length of v⃗, then
where N is a number (scalar).
If we have a vector u⃗ and we want to obtain a new vector v⃗ which has the same direction but is N-times smaller, we can write the equation in two ways:
When multiplying a 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, we obtain the equation |v⃗| = -2 × |u⃗| (the vector v⃗ is twice as long as the vector u⃗ 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:
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.
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