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Using Scalar Multiplication with Vectors

Multiplying a vector by a scalar is called scalar multiplication. To perform scalar multiplication, you need to multiply the scalar by each component of the vector.

A scalar is just a fancy word for a real number. The name arises because a scalar scales a vector — that is, it changes the scale of a vector. For example, the real number 2 scales the vector v by a factor of 2 so that 2v is twice as long as v.

Here’s how you multiply the vector


For example, you multiply the vector


by the scalars 2, –4, and 1/3 as follows:


When you multiply a vector by a scalar, the result is a vector.

Geometrically speaking, scalar multiplication achieves the following:

  • Scalar multiplication by a positive number other than 1 changes the magnitude of the vector but not its direction.

  • Scalar multiplication by –1 reverses its direction but doesn’t change its magnitude.

  • Scalar multiplication by any other negative number both reverses the direction of the vector and changes its magnitude.

Scalar multiplication can change the magnitude of a vector by either increasing it or decreasing it.

  • Scalar multiplication by a number greater than 1 or less than –1 increases the magnitude of the vector.

  • Scalar multiplication by a fraction between –1 and 1 decreases the magnitude of the vector.

    Scalar multiplication of a vector changes its magnitude and/or its direction.
    Scalar multiplication of a vector changes its magnitude and/or its direction.

For example, the vector 2p is twice as long as p, the vector 1/2 p is half as long as p, and the vector –p is the same length as p but extends in the opposite direction from the origin (as shown here).

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