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Scalar Definition: Scalars vs. Vectors (Difference Between Scalar and Vector)

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Scalar Definition / Difference Between Scalar and Vector

A scalar is a single real number that is used to measure magnitude (size). Scalars have magnitude or a numerical value, and not direction. They are the opposite of vectors, which have both a magnitude (or numerical value) and a direction. Real numbers like -0.1, .001, 1, 1.435, √13 are always scalars in matrix and linear algebra**; they cannot be vectors without a direction.
Examples of scalars: 10 m; 100 degrees F; 222 K/cal
Examples of vectors: 5 m/sec, North; 10 miles, West;
scalar vs vector

**In physics, some real numbers can be vectors. See “Can a Scalar be Negative?” below.

Scalar and Scalar Multiples in Linear Algebra

Scalars are used in matrix multiplication. When a matrix is multiplied by a number (a scalar), each element in the matrix is multiplied by that number to create a new matrix. In the following image, the matrix {9,3; 5,7} is multiplied by the scalar 2. The new matrix is called a scalar multiple. In this case, it’s a scalar multiple of 2.
scalar definition

Can a Scalar be Negative?

A number like -10 can be a scalar or a vector depending on what situation you are using it in. In linear algebra, scalars can be negative. A negative scalar like -10 would result in a vector in the opposite direction. In physics, scalars and vectors are defined by what happens to them during rotations. Direction is sometimes denoted with a + or – to mean the positive or negative direction relative to a reference point. In this situation, -10 would not be a vector as -10 would mean 10 units in the negative direction from the reference point. To avoid confusion, the word “scalar” in physics is sometimes limited to complex numbers.

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