# Define Scalar

In math, the scalar defines the real numbers and it is used in linear algebra. In vector space, the scalar is related to the vector. It is referred as numerical quantity. The scalar product is performed in vector space. The scalar part is representing the real component of vector. Now we are going to define about scalar.

## Explanation for Define Scalar

Define scalar:

A scalar is a number and it defines the real number in algebra. It is related to vectors and represented through the scalar multiplication. The scalar is an element in vector space.

Define scalar as vector component:

The vector components are called as scalar and fundamental theorem of linear algebra is deriving the scalar values in vector space.

Define scalar product:

The multiplication of two vectors is called as scalar product. The scalar product creates the new vector. It is also called as inner product. For example, `veca` = 2 `veci` + 3 `vecj` + 4 `veck`and `vecb` = 5 `veci` + 6 `vecj` + 7 `veck` . The scalar multiplication is performed as `veca` . `vecb` = (2 `veci` + 3 `vecj` + 4 `veck` ) . (5 `veci` + 6 `vecj ` + 7` veck` ) = (2.3) + (3.6) + (4.7) = 6 + 18 + 28 = 52.

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## More about Define Scalar

Example problems for define scalar:

Problem 1: Find out the scalar multiplication value of vectors `veca` = -5 `veci` - 2 `vecj ` + 3 `veck` and `vecb` = 2 `veci` + 7` vecj` + 4 `veck` .

Solution:

The given vectors are `veca` = -5 `veci` - 2 `vecj` + 3 `veck` and `vecb` = 2 `veci` + 7 `vecj` + 4 `veck` .

Scalar multiplication is performed as,

`veca` . `vecb` = (-5 `veci` - 2 `vecj` + 3 `veck` ) . (2 `veci` + 7 `vecj` + 4 `veck` )

= -10 -14 +12 = -12.

Therefore, the scalar multiplication value is -12.

Problem 2: Find out the scalar multiplication value of vectors `veca` = 4 `veci` - 7 `vecj` + 2 `veck` and `vecb` = 3 `veci` + 4 `vecj` + 1`veck` .

Solution:

The given vectors are `veca` = 4 `veci` - 7 `vecj` + 2 `veck` and `vecb` = 3 `veci` + 4 `vecj` + 1 `veck` .

Scalar multiplication is performed as,

`veca` . `vecb` = (4 `veci` - 7 `vecj` + 2 `veck` ) . (3 `veci` + 4 `vecj` + 1 `veck` )

= 12 - 28 + 2 = -14..

Therefore, the scalar multiplication value is -14.

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Exercise problems for define scalar:

1. Perform the scalar multiplication in given vectors.

`veca ` = 1 `veci` + 3 `vecj` + 5 `veck` and `vecb` = 4 `veci` + 2 `vecj` + 8 `veck`

Solution: The scalar multiplication value is 50.

2. Perform the scalar multiplication in given vectors.

`veca` = 5 `veci` + 2 `vecj ` + 9 `veck` and `vecb` = 2 `veci` + 3 `vecj` + 4 `veck`

Solution: The scalar multiplication value is 52.

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