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How does multiplying a vector by a scalar change its length and direction?
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linear algebra
One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.
In linear algebra, multiplying a vector by a scalar affects its magnitude and orientation as follows:
- Length: The magnitude of the resulting vector is the product of the absolute value of the scalar and the magnitude of the original vector. If the scalar has an absolute value greater than 1, the vector becomes longer; if it is between 0 and 1, the vector becomes shorter.12
- Direction:
- If the scalar is positive, the resulting vector points in the same direction as the original vector.12
- If the scalar is negative, the direction of the vector is reversed (pointed in the opposite direction).12
For example, multiplying a vector by 2 results in a vector twice as long in the same direction, while multiplying by β2 results in a vector twice as long pointing in the opposite direction.13